Probability And Measure

• B1.13

  Probability and Measure

  Part II, 2001

  State and prove Hölder's Inequality.

  [Jensen's inequality, and other standard results, may be assumed.]

  Let $\left(X_{n}\right)$ be a sequence of random variables bounded in $L_{p}$ for some $p>1$. Prove that $\left(X_{n}\right)$ is uniformly integrable.

  Suppose that $X \in L_{p}(\Omega, \mathcal{F}, \mathbb{P})$ for some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and some $p \in(1, \infty)$. Show that $X \in L_{r}(\Omega, \mathcal{F}, \mathbb{P})$ for all $1 \leqslant r<p$ and that $\|X\|_{r}$ is an increasing function of $r$ on $[1, p]$.

  Show further that $\lim _{r \rightarrow 1^{+}}\|X\|_{r}=\|X\|_{1}$.

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• B2.12

  Probability and Measure

  Part II, 2001

  (a) Let $\Omega=(0,1), \mathcal{F}=\mathcal{B}((0,1))$ be the Borel $\sigma$-field and let $\mathbb{P}$ be Lebesgue measure on $(\Omega, \mathcal{F})$. What is the distribution of the random variable $Z$, where $Z(\omega)=2 \omega-1$ ?

  Let $\omega=\sum_{n=1}^{\infty} 2^{-n} R_{n}(\omega)$ be the binary expansion of the point $\omega \in \Omega$ and set $U(\omega)=\sum_{n \text { odd }} 2^{-n} Q_{n}(\omega)$, where $Q_{n}(\omega)=2 R_{n}(\omega)-1$. Find a random variable $V$ independent of $U$ such that $U$ and $V$ are identically distributed and $U+\frac{1}{2} V$ is uniformly distributed on $(-1,1)$.

  (b) Now suppose that on some probability triple $X$ and $Y$ are independent, identicallydistributed random variables such that $X+\frac{1}{2} Y$ is uniformly distributed on $(-1,1)$.

  Let $\phi$ be the characteristic function of $X$. Calculate $\phi(t) / \phi(t / 4)$. Show that the distribution of $X$ must be the same as the distribution of the random variable $U$ in (a).

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• B3.12

  Probability and Measure

  Part II, 2001

  State and prove Birkhoff's almost-everywhere ergodic theorem.

  [You need not prove convergence in $L_{p}$ and the maximal ergodic lemma may be assumed provided that it is clearly stated.]

  Let $\Omega=[0,1), \mathcal{F}=\mathcal{B}([0,1))$ be the Borel $\sigma$-field and let $\mathbb{P}$ be Lebesgue measure on $(\Omega, \mathcal{F})$. Give an example of an ergodic measure-preserving map $\theta: \Omega \rightarrow \Omega$ (you need not prove it is ergodic).

  Let $f(x)=x$ for $x \in[0,1)$. Find (at least for all $x$ outside a set of measure zero)

  $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n}\left(f \circ \theta^{i-1}\right)(x) .$$

  Briefly justify your answer.

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• B4.11

  Probability and Measure

  Part II, 2001

  State the first and second Borel-Cantelli Lemmas and the Kolmogorov 0-1 law.

  Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of independent random variables with distribution given

  by

  $$\mathbb{P}\left(X_{n}=n\right)=\frac{1}{n}=1-\mathbb{P}\left(X_{n}=0\right)$$

  and set $S_{n}=\sum_{i=1}^{n} X_{i}$.

  (a) Show that there exist constants $0 \leqslant c_{1} \leqslant c_{2} \leqslant \infty$ such that $\lim \inf _{n}\left(S_{n} / n\right)=c_{1}$, almost surely and $\lim \sup _{n}\left(S_{n} / n\right)=c_{2}$ almost surely.

  (b) Let $Y_{k}=\sum_{i=k+1}^{2 k} X_{i}$ and $\tilde{Y}_{k}=\sum_{i=1}^{k} Z_{i}^{(k)}$, where $\left(Z_{i}^{(k)}\right)_{i=1}^{k}$ are independent with

  $$\mathbb{P}\left(Z_{i}^{(k)}=k\right)=\frac{1}{2 k}=1-\mathbb{P}\left(Z_{i}^{(k)}=0\right), \quad 1 \leqslant i \leqslant k,$$

  and suppose that $\alpha \in \mathbb{Z}^{+}$.

  Use the fact that $\mathbb{P}\left(Y_{k} \geqslant \alpha k\right) \geqslant \mathbb{P}\left(\tilde{Y}_{k} \geqslant \alpha k\right)$ to show that there exists $p_{\alpha}>0$ such that $\mathbb{P}\left(Y_{k} \geqslant \alpha k\right) \geqslant p_{\alpha}$ for all sufficiently large $k$.

  [You may use the Poisson approximation to the binomial distribution without proof.]

  By considering a suitable subsequence of $\left(Y_{k}\right)$, or otherwise, show that $c_{2}=\infty$.

  (c) Show that $c_{1} \leqslant 1$. Consider an appropriately chosen sequence of random times $T_{i}$, with $2 T_{i} \leqslant T_{i+1}$, for which $\left(S_{T_{i}} / T_{i}\right) \leqslant 3 c_{1} / 2$. Using the fact that the random variables $\left(Y_{T_{i}}\right)$ are independent, and by considering the events $\left\{Y_{T_{i}}=0\right\}$, or otherwise, show that $c_{1}=0$.

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• B1.13

  Probability and Measure

  Part II, 2002

  State and prove Dynkin's $\pi$-system lemma.

  Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\left(A_{n}\right)$ be a sequence of independent events such that $\lim _{n \rightarrow \infty} \mathbb{P}\left(A_{n}\right)=p$. Let $\mathcal{G}=\sigma\left(A_{1}, A_{2}, \ldots\right)$. Prove that

  $$\lim _{n \rightarrow \infty} \mathbb{P}\left(G \cap A_{n}\right)=p \mathbb{P}(G)$$

  for all $G \in \mathcal{G}$.

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• B2.12

  Probability and Measure

  Part II, 2002

  Let $\left(X_{n}\right)$ be a sequence of non-negative random variables on a common probability space with $\mathbb{E} X_{n} \leqslant 1$, such that $X_{n} \rightarrow 0$ almost surely. Determine which of the following statements are necessarily true, justifying your answers carefully: (a) $\mathbb{P}\left(X_{n} \geqslant 1\right) \rightarrow 0$ as $n \rightarrow \infty$; (b) $\mathbb{E} X_{n} \rightarrow 0$ as $n \rightarrow \infty$; (c) $\mathbb{E}\left(\sin \left(X_{n}\right)\right) \rightarrow 0$ as $n \rightarrow \infty$; (d) $\mathbb{E}\left(\sqrt{X_{n}}\right) \rightarrow 0$ as $n \rightarrow \infty$.

  [Standard limit theorems for integrals, and results about uniform integrability, may be used without proof provided that they are clearly stated.]

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• B3.12

  Probability and Measure

  Part II, 2002

  Derive the characteristic function of a real-valued random variable which is normally distributed with mean $\mu$ and variance $\sigma^{2}$. What does it mean to say that an $\mathbb{R}^{n}$-valued random variable has a multivariate Gaussian distribution? Prove that the distribution of such a random variable is determined by its $\left(\mathbb{R}^{n}\right.$-valued) mean and its covariance matrix.

  Let $X$ and $Y$ be random variables defined on the same probability space such that $(X, Y)$ has a Gaussian distribution. Show that $X$ and $Y$ are independent if and only if $\operatorname{cov}(X, Y)=0$. Show that, even if they are not independent, one may always write $X=a Y+Z$ for some constant $a$ and some random variable $Z$ independent of $Y$.

  [The inversion theorem for characteristic functions and standard results about independence may be assumed.]

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• B4.11

  Probability and Measure

  Part II, 2002

  State Birkhoff's Almost Everywhere Ergodic Theorem for measure-preserving transformations. Define what it means for a sequence of random variables to be stationary. Explain briefly how the stationarity of a sequence of random variables implies that a particular transformation is measure-preserving.

  A bag contains one white ball and one black ball. At each stage of a process one ball is picked from the bag (uniformly at random) and then returned to the bag together with another ball of the same colour. Let $X_{n}$ be a random variable which takes the value 0 if the $n$th ball added to the bag is white and 1 if it is black.

  (a) Show that the sequence $X_{1}, X_{2}, X_{3}, \ldots$ is stationary and hence that the proportion of black balls in the bag converges almost surely to some random variable $R$.

  (b) Find the distribution of $R$.

  [The fact that almost-sure convergence implies convergence in distribution may be used without proof.]

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• B1.13

  Probability and Measure

  Part II, 2003

  State and prove the first Borel-Cantelli Lemma.

  Suppose that $\left(F_{n}\right)$ is a sequence of events in a common probability space such that $\mathbb{P}\left(F_{i} \cap F_{j}\right) \leqslant \mathbb{P}\left(F_{i}\right) \cdot \mathbb{P}\left(F_{j}\right)$ whenever $i \neq j$ and that $\sum_{n} \mathbb{P}\left(F_{n}\right)=\infty$.

  Let $1_{F_{n}}$ be the indicator function of $F_{n}$ and let

  $$S_{n}=\sum_{k \leqslant n} 1_{F_{k}} ; \mu_{n}=\mathbb{E}\left(S_{n}\right)$$

  Use Chebyshev's inequality to show that

  $$\mathbb{P}\left(S_{n}<\frac{1}{2} \mu_{n}\right) \leqslant \mathbb{P}\left(\left|S_{n}-\mu_{n}\right|>\frac{1}{2} \mu_{n}\right) \leqslant \frac{4}{\mu_{n}}$$

  Deduce, using the first Borel-Cantelli Lemma, that $\mathbb{P}\left(F_{n}\right.$ infinitely often $)=1$.

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• B2.12

  Probability and Measure

  Part II, 2003

  Let $H$ be a Hilbert space and let $V$ be a closed subspace of $H$. Let $x \in H$. Show that there is a unique decomposition $x=u+v$ such that $v \in V$ and $u \in V^{\perp}$.

  Now suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space and let $X \in L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. Suppose $\mathcal{G}$ is a sub- $\sigma$-algebra of $\mathcal{F}$. Define $\mathbb{E}(X \mid \mathcal{G})$ using a decomposition of the above type. Show that $\mathbb{E}\left(\mathbb{E}(X \mid \mathcal{G}) .1_{A}\right)=\mathbb{E}\left(X .1_{A}\right)$ for each set $A \in \mathcal{G}$.

  Let $\mathcal{G}_{1} \subseteq \mathcal{G}_{2}$ be two sub- $\sigma$-algebras of $\mathcal{F}$. Show that (a) $\mathbb{E}\left(\mathbb{E}\left(X \mid \mathcal{G}_{1}\right) \mid \mathcal{G}_{2}\right)=\mathbb{E}\left(X \mid \mathcal{G}_{1}\right)$; (b) $\mathbb{E}\left(\mathbb{E}\left(X \mid \mathcal{G}_{2}\right) \mid \mathcal{G}_{1}\right)=\mathbb{E}\left(X \mid \mathcal{G}_{1}\right)$.

  No general theorems about projections on Hilbert spaces may be quoted without proof.

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• B3.12

  Probability and Measure

  Part II, 2003

  Explain what is meant by the characteristic function $\phi$ of a real-valued random variable and prove that $|\phi|^{2}$ is also a characteristic function of some random variable.

  Let us say that a characteristic function $\phi$ is infinitely divisible when, for each $n \geqslant 1$, we can write $\phi=\left(\phi_{n}\right)^{n}$ for some characteristic function $\phi_{n}$. Prove that, in this case, the limit

  $$\psi(t)=\lim _{n \rightarrow \infty}\left|\phi_{2 n}(t)\right|^{2}$$

  exists for all real $t$ and is continuous at $t=0$.

  Using Lévy's continuity theorem for characteristic functions, which you should state carefully, deduce that $\psi$ is a characteristic function. Hence show that, if $\phi$ is infinitely divisible, then $\phi(t)$ cannot vanish for any real $t$.

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• B4.11

  Probability and Measure

  Part II, 2003

  Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable with respect to Lebesgue measure $\mu$ on $[a, b]$. Prove that, if

  $$\int_{J} f d \mu=0$$

  for every sub-interval $J$ of $[a, b]$, then $f=0$ almost everywhere on $[a, b]$.

  Now define

  $$F(x)=\int_{a}^{x} f d \mu .$$

  Prove that $F$ is continuous on $[a, b]$. Show that, if $F$ is zero on $[a, b]$, then $f$ is zero almost everywhere on $[a, b]$.

  Suppose now that $f$ is bounded and Lebesgue integrable on $[a, b]$. By applying the Dominated Convergence Theorem to

  $$F_{n}(x)=\frac{F\left(x+\frac{1}{n}\right)-F(x)}{\frac{1}{n}},$$

  or otherwise, show that, if $F$ is differentiable on $[a, b]$, then $F^{\prime}=f$ almost everywhere on $[a, b]$.

  The functions $f_{n}:[a, b] \rightarrow \mathbb{R}$ have the properties:

  (a) $f_{n}$ converges pointwise to a differentiable function $g$ on $[a, b]$,

  (b) each $f_{n}$ has a continuous derivative $f_{n}^{\prime}$ with $\left|f_{n}^{\prime}(x)\right| \leqslant 1$ on $[a, b]$,

  (c) $f_{n}^{\prime}$ converges pointwise to some function $h$ on $[a, b]$.

  Deduce that

  $$h(x)=\lim _{n \rightarrow \infty}\left(\frac{d f_{n}(x)}{d x}\right)=\frac{d}{d x}\left(\lim _{n \rightarrow \infty} f_{n}(x)\right)=g^{\prime}(x)$$

  almost everywhere on $[a, b]$.

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• B1.13

  Probability and Measure

  Part II, 2004

  Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{A}=\left(A_{i}: i=1,2, \ldots\right)$ be a sequence of events.

  (a) What is meant by saying that $\mathcal{A}$ is a family of independent events?

  (b) Define the events $\left\{A_{n}\right.$ infinitely often $\}$ and $\left\{A_{n}\right.$ eventually $\}$. State and prove the two Borel-Cantelli lemmas for $\mathcal{A}$.

  (c) Let $X_{1}, X_{2}, \ldots$ be the outcomes of a sequence of independent flips of a fair coin,

  $$\mathbb{P}\left(X_{i}=0\right)=\mathbb{P}\left(X_{i}=1\right)=\frac{1}{2} \quad \text { for } i \geqslant 1$$

  Let $L_{n}$ be the length of the run beginning at the $n^{\text {th }}$flip. For example, if the first fourteen outcomes are 01110010000110 , then $L_{1}=1, L_{2}=3, L_{3}=2$, etc.

  Show that

  $$\mathbb{P}\left(\limsup _{n \rightarrow \infty} \frac{L_{n}}{\log _{2} n}>1\right)=0$$

  and furthermore that

  $$\mathbb{P}\left(\limsup _{n \rightarrow \infty} \frac{L_{n}}{\log _{2} n}=1\right)=1$$

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• B2.12

  Probability and Measure

  Part II, 2004

  Let $(\Omega, \mathcal{F}, \mu)$ be a measure space and let $1 \leqslant p \leqslant \infty$.

  (a) Define the $L^{p}$-norm $\|f\|_{p}$ of a measurable function $f: \Omega \rightarrow \mathbb{R}$, and define the space $L^{p}(\Omega, \mathcal{F}, \mu) .$

  (b) Prove Minkowski's inequality:

  $$\|f+g\|_{p} \leqslant\|f\|_{p}+\|g\|_{p} \text { for } f, g \in L^{p}(\Omega, \mathcal{F}, \mu), 1 \leqslant p \leqslant \infty$$

  [You may use Hölder's inequality without proof provided it is clearly stated.]

  (c) Explain what is meant by saying that $L^{p}(\Omega, \mathcal{F}, \mu)$ is complete. Show that $L^{\infty}(\Omega, \mathcal{F}, \mu)$ is complete.

  (d) Suppose that $\left\{f_{n}: n \geqslant 1\right\}$ is a sequence of measurable functions satisfying $\left\|f_{n}\right\|_{p} \rightarrow 0$ as $n \rightarrow \infty$.

  (i) Show that if $p=\infty$, then $f_{n} \rightarrow 0$ almost everywhere.

  (ii) When $1 \leqslant p<\infty$, give an example of a measure space $(\Omega, \mathcal{F}, \mu)$ and such a sequence $\left\{f_{n}\right\}$ such that, for all $\omega \in \Omega, f_{n}(\omega) \nrightarrow 0$ as $n \rightarrow \infty$.

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• B3.12

  Probability and Measure

  Part II, 2004

  (a) Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\theta: \Omega \rightarrow \Omega$ be measurable. What is meant by saying that $\theta$ is measure-preserving? Define an invariant event and an invariant random variable, and explain what is meant by saying that $\theta$ is ergodic.

  (b) Let $m$ be a probability measure on $(\mathbb{R}, \mathcal{B})$. Let

  $$\Omega=\mathbb{R}^{\mathbb{N}}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{i} \in \mathbb{R} \text { for } i \geqslant 1\right\}$$

  let $\mathcal{F}$ be the smallest $\sigma$-field of $\Omega$ with respect to which the coordinate maps $X_{n}(x)=x_{n}$, for $x \in \Omega, n \geqslant 1$, are measurable, and let $\mathbb{P}$ be the unique probability measure on $(\Omega, \mathcal{F})$ satisfying

  $$\mathbb{P}\left(X_{i} \in A_{i} \text { for } 1 \leqslant i \leqslant n\right)=\prod_{i=1}^{n} m\left(A_{i}\right)$$

  for all $A_{i} \in \mathcal{B}, n \geqslant 1$. Define $\theta: \Omega \rightarrow \Omega$ by $\theta(x)=\left(x_{2}, x_{3}, \ldots\right)$ for $x=\left(x_{1}, x_{2}, \ldots\right)$.

  (i) Show that $\theta$ is measurable and measure-preserving.

  (ii) Define the tail $\sigma$-field $\mathcal{T}$ of the coordinate maps $X_{1}, X_{2}, \ldots$, and show that the invariant $\sigma$-field $\mathcal{I}$ of $\theta$ satisfies $\mathcal{I} \subseteq \mathcal{T}$. Deduce that $\theta$ is ergodic. [Any general result used must be stated clearly but the proof may be omitted.]

  (c) State Birkhoff's ergodic theorem and explain how to deduce that, given independent identically-distributed integrable random variables $Y_{1}, Y_{2}, \ldots$, there exists $\nu \in \mathbb{R}$ such that

  $$\frac{1}{n}\left(Y_{1}+Y_{2}+\cdots+Y_{n}\right) \rightarrow \nu \quad \text { a.e. as } \quad n \rightarrow \infty .$$

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• B4.11

  Probability and Measure

  Part II, 2004

  Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $X, X_{1}, X_{2}, \ldots$ be random variables. Write an essay in which you discuss the statement: if $X_{n} \rightarrow X$ almost everywhere, then $\mathbb{E}\left(X_{n}\right) \rightarrow \mathbb{E}(X)$. You should include accounts of monotone, dominated, and bounded convergence, and of Fatou's lemma.

  [You may assume without proof the following fact. Let $(\Omega, \mathcal{F}, \mu)$ be a measure space, and let $f: \Omega \rightarrow \mathbb{R}$ be non-negative with finite integral $\mu(f) .$ If $\left(f_{n}: n \geqslant 1\right)$ are non-negative measurable functions with $f_{n}(\omega) \uparrow f(\omega)$ for all $\omega \in \Omega$, then $\mu\left(f_{n}\right) \rightarrow \mu(f)$ as $n \rightarrow \infty$.]

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• 1.II $. 25 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2005

  Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. For $\mathcal{G} \subseteq \mathcal{F}$, what is meant by saying that $\mathcal{G}$ is a $\pi$-system? State the 'uniqueness of extension' theorem for measures on $\sigma(\mathcal{G})$ having given values on $\mathcal{G}$.

  For $\mathcal{G}, \mathcal{H} \subseteq \mathcal{F}$, we call $\mathcal{G}, \mathcal{H}$ independent if

  $$\mathbb{P}(G \cap H)=\mathbb{P}(G) \mathbb{P}(H) \quad \text { for all } \quad G \in \mathcal{G}, H \in \mathcal{H}$$

  If $\mathcal{G}$ and $\mathcal{H}$ are independent $\pi$-systems, show that $\sigma(\mathcal{G})$ and $\sigma(\mathcal{H})$ are independent.

  Let $Y_{1}, Y_{2}, \ldots, Y_{m}, Z_{1}, Z_{2}, \ldots, Z_{n}$ be independent random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Show that the $\sigma$-fields $\sigma(Y)=\sigma\left(Y_{1}, Y_{2}, \ldots, Y_{m}\right)$ and $\sigma(Z)=\sigma\left(Z_{1}, Z_{2}, \ldots, Z_{n}\right)$ are independent.

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• 2.II.25J

  Probability and Measure

  Part II, 2005

  Let $\mathcal{R}$ be a family of random variables on the common probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What is meant by saying that $\mathcal{R}$ is uniformly integrable? Explain the use of uniform integrability in the study of convergence in probability and in $L^{1}$. [Clear definitions should be given of any terms used, but proofs may be omitted.]

  Let $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ be uniformly integrable families of random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Show that the family $\mathcal{R}$ given by

  $$\mathcal{R}=\left\{X+Y: X \in \mathcal{R}_{1}, Y \in \mathcal{R}_{2}\right\}$$

  is uniformly integrable.

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• $3 . \mathrm{II} . 24 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2005

  Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. For a measurable function $f: \Omega \rightarrow \mathbb{R}$, and $p \in[1, \infty)$, let $\|f\|_{p}=\left[\mu\left(|f|^{p}\right)\right]^{1 / p}$. Let $L^{p}$ be the space of all such $f$ with $\|f\|_{p}<\infty$. Explain what is meant by each of the following statements:

  (a) A sequence of functions $\left(f_{n}: n \geqslant 1\right)$ is Cauchy in $L^{p}$.

  (b) $L^{p}$ is complete.

  Show that $L^{p}$ is complete for $p \in[1, \infty)$.

  Take $\Omega=(1, \infty), \mathcal{F}$ the Borel $\sigma$-field of $\Omega$, and $\mu$ the Lebesgue measure on $(\Omega, \mathcal{F})$. For $p=1,2$, determine which if any of the following sequences of functions are Cauchy in $L^{p}$

  (i) $f_{n}(x)=x^{-1} 1_{(1, n)}(x)$,

  (ii) $g_{n}(x)=x^{-2} 1_{(1, n)}(x)$,

  where $1_{A}$ denotes the indicator function of the set $A$.

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• 4.II.25J

  Probability and Measure

  Part II, 2005

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be Borel-measurable. State Fubini's theorem for the double integral

  $$\int_{y \in \mathbb{R}} \int_{x \in \mathbb{R}} f(x, y) d x d y$$

  Let $0<a<b$. Show that the function

  $$f(x, y)= \begin{cases}e^{-x y} & \text { if } x \in(0, \infty), y \in[a, b] \\ 0 & \text { otherwise }\end{cases}$$

  is measurable and integrable on $\mathbb{R}^{2}$.

  Evaluate

  $$\int_{0}^{\infty} \frac{e^{-a x}-e^{-b x}}{x} d x$$

  by Fubini's theorem or otherwise.

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• 1.II $. 25 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2006

  Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of (real-valued, Borel-measurable) random variables on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$.

  (a) Let $\left(A_{n}\right)_{n \in \mathbb{N}}$ be a sequence of events in $\mathcal{A}$.

  What does it mean for the events $\left(A_{n}\right)_{n \in \mathbb{N}}$ to be independent?

  What does it mean for the random variables $\left(X_{n}\right)_{n \in \mathbb{N}}$ to be independent?

  (b) Define the tail $\sigma$-algebra $\mathcal{T}$ for a sequence $\left(X_{n}\right)_{n \in \mathbb{N}}$ and state Kolmogorov's $0-1$ law.

  (c) Consider the following events in $\mathcal{A}$,

  $$\begin{gathered} \left\{X_{n} \leqslant 0 \text { eventually }\right\} \\ \left\{\lim _{n \rightarrow \infty} X_{1}+\ldots+X_{n} \text { exists }\right\} \\ \left\{X_{1}+\ldots+X_{n} \leqslant 0 \text { infinitely often }\right\} \end{gathered}$$

  Which of them are tail events for $\left(X_{n}\right)_{n \in \mathbb{N}}$ ? Justify your answers.

  (d) Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be independent random variables with

  $$\mathbb{P}\left(X_{n}=0\right)=\mathbb{P}\left(X_{n}=1\right)=\frac{1}{2} \quad \text { for all } n \in \mathbb{N},$$

  and define $U_{n}=X_{1} X_{2}+X_{2} X_{3}+\ldots+X_{2 n} X_{2 n+1}$.

  Show that $U_{n} / n \rightarrow c$ a.s. for some $c \in \mathbb{R}$, and determine $c$.

  [Standard results may be used without proof, but should be clearly stated.]

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• 2.II $. 25 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2006

  (a) What is meant by saying that $(\Omega, \mathcal{A}, \mu)$ is a measure space? Your answer should include clear definitions of any terms used.

  (b) Consider the following sequence of Borel-measurable functions on the measure space $(\mathbb{R}, \mathcal{L}, \lambda)$, with the Lebesgue $\sigma$-algebra $\mathcal{L}$ and Lebesgue measure $\lambda$ :

  $$f_{n}(x)=\left\{\begin{array}{ll} 1 / n & \text { if } 0 \leqslant x \leqslant e^{n} ; \\ 0 & \text { otherwise } \end{array} \quad \text { for } n \in \mathbb{N}\right.$$

  For each $p \in[1, \infty]$, decide whether the sequence $\left(f_{n}\right)_{n \in \mathbb{N}}$ converges in $L^{p}$ as $n \rightarrow \infty$.

  Does $\left(f_{n}\right)_{n \in \mathbb{N}}$ converge almost everywhere?

  Does $\left(f_{n}\right)_{n \in \mathbb{N}}$ converge in measure?

  Justify your answers.

  For parts (c) and (d), let $\left(f_{n}\right)_{n \in \mathbb{N}}$ be a sequence of real-valued, Borel-measurable functions on a probability space $(\Omega, \mathcal{A}, \mu)$.

  (c) Prove that $\left\{x \in \Omega: f_{n}(x)\right.$ converges to a finite limit $\} \in \mathcal{A}$.

  (d) Show that $f_{n} \rightarrow 0$ almost surely if and only if $\sup _{m \geqslant n}\left|f_{m}\right| \rightarrow 0$ in probability.

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• $3 . \mathrm{II} . 24 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2006

  Let $X$ be a real-valued random variable. Define the characteristic function $\phi_{X}$. Show that $\phi_{X}(u) \in \mathbb{R}$ for all $u \in \mathbb{R}$ if and only if $X$ and $-X$ have the same distribution.

  For parts (a) and (b) below, let $X$ and $Y$ be independent and identically distributed random variables.

  (a) Show that $X=Y$ almost surely implies that $X$ is almost surely constant.

  (b) Suppose that there exists $\varepsilon>0$ such that $\left|\phi_{X}(u)\right|=1$ for all $|u|<\varepsilon$. Calculate $\phi_{X-Y}$ to show that $\mathbb{E}(1-\cos (u(X-Y)))=0$ for all $|u|<\varepsilon$, and conclude that $X$ is almost surely constant.

  (c) Let $X, Y$, and $Z$ be independent $\mathrm{N}(0,1)$ random variables. Calculate the characteristic function of $\eta=X Y-Z$, given that $\phi_{X}(u)=e^{-u^{2} / 2}$.

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• 4.II $. 25 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2006

  Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $f: \Omega \rightarrow \mathbb{R}$ a measurable function.

  (a) Explain what is meant by saying that $f$ is integrable, and how the integral $\int_{\Omega} f d \mu$ is defined, starting with integrals of $\mathcal{A}$-simple functions.

  [Your answer should consist of clear definitions, including the ones for $\mathcal{A}$-simple functions and their integrals.]

  (b) For $f: \Omega \rightarrow[0, \infty)$ give a specific sequence $\left(g_{n}\right)_{n \in \mathbb{N}}$ of $\mathcal{A}$-simple functions such that $0 \leqslant g_{n} \leqslant f$ and $g_{n}(x) \rightarrow f(x)$ for all $x \in \Omega$. Justify your answer.

  (c) Suppose that that $\mu(\Omega)<\infty$ and let $f_{1}, f_{2}, \ldots: \Omega \rightarrow \mathbb{R}$ be measurable functions such that $f_{n}(x) \rightarrow 0$ for all $x \in \Omega$. Prove that, if

  $$\lim _{c \rightarrow \infty} \sup _{n \in \mathbb{N}} \int_{\left|f_{n}\right|>c}\left|f_{n}\right| d \mu=0$$

  then $\int_{\Omega} f_{n} d \mu \rightarrow 0$.

  Give an example with $\mu(\Omega)<\infty$ such that $f_{n}(x) \rightarrow 0$ for all $x \in \Omega$, but $\int_{\Omega} f_{n} d \mu \nrightarrow 0$, and justify your answer.

  (d) State and prove Fatou's Lemma for a sequence of non-negative measurable functions.

  [Standard results on measurability and integration may be used without proof.]

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• 1.II.25J

  Probability and Measure

  Part II, 2007

  Let $E$ be a set and $\mathcal{E} \subseteq \mathcal{P}(E)$ be a set system.

  (a) Explain what is meant by a $\pi$-system, a $d$-system and a $\sigma$-algebra.

  (b) Show that $\mathcal{E}$ is a $\sigma$-algebra if and only if $\mathcal{E}$ is a $\pi$-system and a $d$-system.

  (c) Which of the following set systems $\mathcal{E}_{1}, \mathcal{E}_{2}, \mathcal{E}_{3}$ are $\pi$-systems, $d$-systems or $\sigma$-algebras? Justify your answers. ( $\#(A)$ denotes the number of elements in $A$.)

  $E_{1}=\{1,2, \ldots, 10\}$ and $\mathcal{E}_{1}=\left\{A \subseteq E_{1}: \#(A)\right.$ is even $\}$,

  $E_{2}=\mathbb{N}=\{1,2, \ldots\}$ and $\mathcal{E}_{2}=\left\{A \subseteq E_{2}: \#(A)\right.$ is even or $\left.\#(A)=\infty\right\}$,

  $E_{3}=\mathbb{R}$ and $\mathcal{E}_{3}=\{(a, b): a, b \in \mathbb{R}, a<b\} \cup\{\emptyset\} .$

  (d) State and prove the theorem on the uniqueness of extension of a measure.

  [You may use standard results from the lectures without proof, provided they are clearly stated.]

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• 2.II.25J

  Probability and Measure

  Part II, 2007

  (a) State and prove the first Borel-Cantelli lemma. State the second Borel-Cantelli lemma.

  (b) Let $X_{1}, X_{2}, \ldots$ be a sequence of independent random variables that converges in probability to the limit $X$. Show that $X$ is almost surely constant.

  A sequence $X_{1}, X_{2}, \ldots$ of random variables is said to be completely convergent to $X$ if

  $$\sum_{n \in \mathbb{N}} \mathbb{P}\left(A_{n}(\epsilon)\right)<\infty \quad \text { for all } \epsilon>0, \quad \text { where } A_{n}(\epsilon)=\left\{\left|X_{n}-X\right|>\epsilon\right\}$$

  (c) Show that complete convergence implies almost sure convergence.

  (d) Show that, for sequences of independent random variables, almost sure convergence also implies complete convergence.

  (e) Find a sequence of (dependent) random variables that converges almost surely but does not converge completely.

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• 3.II.24J

  Probability and Measure

  Part II, 2007

  Let $(E, \mathcal{E}, \mu)$ be a finite measure space, i.e. $\mu(E)<\infty$, and let $1 \leqslant p \leqslant \infty$.

  (a) Define the $L^{p}$-norm $\|f\|_{p}$ of a measurable function $f: E \rightarrow \overline{\mathbb{R}}$, define the space $L^{p}(E, \mathcal{E}, \mu)$ and define convergence in $L^{p} .$

  In the following you may use inequalities from the lectures without proof, provided they are clearly stated.

  (b) Let $f, f_{1}, f_{2}, \ldots \in L^{p}(E, \mathcal{E}, \mu)$. Show that $f_{n} \rightarrow f$ in $L^{p}$ implies $\left\|f_{n}\right\|_{p} \rightarrow\|f\|_{p}$.

  (c) Let $f: E \rightarrow \mathbb{R}$ be a bounded measurable function with $\|f\|_{\infty}>0$. Let

  $$M_{n}=\int_{E}|f|^{n} d \mu$$

  Show that $M_{n} \in(0, \infty)$ and $M_{n+1} M_{n-1} \geqslant M_{n}^{2}$.

  By using Jensen's inequality, or otherwise, show that

  $$\mu(E)^{-1 / n}\|f\|_{n} \leqslant M_{n+1} / M_{n} \leqslant\|f\|_{\infty}$$

  Prove that $\lim _{n \rightarrow \infty} M_{n+1} / M_{n}=\|f\|_{\infty} .$

  $\left[\right.$ Observe that $\left.|f| \geqslant 1_{\left\{|f|>\|f\|_{\infty}-\epsilon\right\}}\left(\|f\|_{\infty}-\epsilon\right) .\right]$

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• 4.II.25J

  Probability and Measure

  Part II, 2007

  Let $(E, \mathcal{E}, \mu)$ be a measure space with $\mu(E)<\infty$ and let $\theta: E \rightarrow E$ be measurable.

  (a) Define an invariant set $A \in \mathcal{E}$ and an invariant function $f: E \rightarrow \mathbb{R}$.

  What is meant by saying that $\theta$ is measure-preserving?

  What is meant by saying that $\theta$ is ergodic?

  (b) Which of the following functions $\theta_{1}$ to $\theta_{4}$ is ergodic? Justify your answer.

  On the measure space $([0,1], \mathcal{B}([0,1]), \mu)$ with Lebesgue measure $\mu$ consider

  $$\theta_{1}(x)=1+x, \quad \theta_{2}(x)=x^{2}, \quad \theta_{3}(x)=1-x$$

  On the discrete measure space $\left(\{-1,1\}, \mathcal{P}(\{-1,1\}), \frac{1}{2} \delta_{-1}+\frac{1}{2} \delta_{1}\right)$ consider

  $$\theta_{4}(x)=-x$$

  (c) State Birkhoff's almost everywhere ergodic theorem.

  (d) Let $\theta$ be measure-preserving and let $f: E \rightarrow \mathbb{R}$ be bounded.

  Prove that $\frac{1}{n}\left(f+f \circ \theta+\ldots+f \circ \theta^{n-1}\right)$ converges in $L^{p}$ for all $p \in[1, \infty)$.

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• 1.II $. 25 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2008

  State the Dominated Convergence Theorem.

  Hence or otherwise prove Kronecker's Lemma: if $\left(a_{j}\right)$ is a sequence of non-negative reals such that

  $$\sum_{j=1}^{\infty} \frac{a_{j}}{j}<\infty$$

  then

  $$n^{-1} \sum_{j=1}^{n} a_{j} \rightarrow 0 \quad(n \rightarrow \infty)$$

  Let $\xi_{1}, \xi_{2}, \ldots$ be independent $N(0,1)$ random variables and set $S_{n}=\xi_{1}+\ldots+\xi_{n}$. Let $\mathcal{F}_{0}$ be the collection of all finite unions of intervals of the form $(a, b)$, where $a$ and $b$ are rational, together with the whole line $\mathbb{R}$. Prove that with probability 1 the limit

  $$m(B) \equiv \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{n} I_{B}\left(S_{j}\right)$$

  exists for all $B \in \mathcal{F}_{0}$, and identify it. Is it possible to extend $m$ defined on $\mathcal{F}_{0}$ to a measure on the Borel $\sigma$-algebra of $\mathbb{R}$ ? Justify your answer.

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• 2.II $. 25 J$

  Probability and Measure

  Part II, 2008

  Explain what is meant by a simple function on a measurable space $(S, \mathcal{S})$.

  Let $(S, \mathcal{S}, \mu)$ be a finite measure space and let $f: S \rightarrow \mathbb{R}$ be a non-negative Borel measurable function. State the definition of the integral of $f$ with respect to $\mu$.

  Prove that, for any sequence of simple functions $\left(g_{n}\right)$ such that $0 \leqslant g_{n}(x) \uparrow f(x)$ for all $x \in S$, we have

  $$\int g_{n} d \mu \uparrow \int f d \mu .$$

  State and prove the Monotone Convergence Theorem for finite measure spaces.

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• $3 . \mathrm{II} . 24 \mathrm{~J} \quad$

  Probability and Measure

  Part II, 2008

  (i) What does it mean to say that a sequence of random variables $\left(X_{n}\right)$ converges in probability to $X$ ? What does it mean to say that the sequence $\left(X_{n}\right)$ converges in distribution to $X$ ? Prove that if $X_{n} \rightarrow X$ in probability, then $X_{n} \rightarrow X$ in distribution.

  (ii) What does it mean to say that a sequence of random variables $\left(X_{n}\right)$ is uniformly integrable? Show that, if $\left(X_{n}\right)$ is uniformly integrable and $X_{n} \rightarrow X$ in distribution, then $\mathbb{E}\left(X_{n}\right) \rightarrow \mathbb{E}(X)$.

  [Standard results from the course may be used without proof if clearly stated.]

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• 4.II $. 25 \mathrm{~J}$

  Probability and Measure

  Part II, 2008

  (i) A stepfunction is any function $s$ on $\mathbb{R}$ which can be written in the form

  $$s(x)=\sum_{k=1}^{n} c_{k} I_{\left(a_{k}, b_{k}\right]}(x), \quad x \in \mathbb{R},$$

  where $a_{k}, b_{k}, c_{k}$ are real numbers, with $a_{k}<b_{k}$ for all $k$. Show that the set of all stepfunctions is dense in $L^{1}(\mathbb{R}, \mathcal{B}, \mu)$. Here, $\mathcal{B}$ denotes the Borel $\sigma$-algebra, and $\mu$ denotes Lebesgue measure.

  [You may use without proof the fact that, for any Borel set $B$ of finite measure, and any $\varepsilon>0$, there exists a finite union of intervals $A$ such that $\mu(A \triangle B)<\varepsilon$.]

  (ii) Show that the Fourier transform

  $$\hat{s}(t)=\int_{\mathbb{R}} s(x) e^{i t x} d x$$

  of a stepfunction has the property that $\hat{s}(t) \rightarrow 0$ as $|t| \rightarrow \infty$.

  (iii) Deduce that the Fourier transform of any integrable function has the same property.

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• Paper 3, Section II, J

  Probability and Measure

  Part II, 2009

  State and prove the first and second Borel-Cantelli lemmas.

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent Cauchy random variables. Thus, each $X_{n}$ is real-valued, with density function

  $$f(x)=\frac{1}{\pi\left(1+x^{2}\right)} .$$

  Show that

  $$\limsup _{n \rightarrow \infty} \frac{\log X_{n}}{\log n}=c, \quad \text { almost surely, }$$

  for some constant $c$, to be determined.

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• Paper 4, Section II, J

  Probability and Measure

  Part II, 2009

  Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G}$ be a sub- $\sigma$-algebra of $\mathcal{F}$. Show that, for any random variable $X \in L^{2}(\mathbb{P})$, there exists a $\mathcal{G}$-measurable random variable $Y \in L^{2}(\mathbb{P})$ such that $\mathbb{E}((X-Y) Z)=0$ for all $\mathcal{G}$-measurable random variables $Z \in L^{2}(\mathbb{P})$.

  [You may assume without proof the completeness of $L^{2}(\mathbb{P}) .$ ]

  Let $(G, X)$ be a Gaussian random variable in $\mathbb{R}^{2}$, with mean $(\mu, \nu)$ and covariance $\operatorname{matrix}\left(\begin{array}{cc}u & v \\ v & w\end{array}\right)$. Assume that $\mathcal{F}=\sigma(G, X)$ and $\mathcal{G}=\sigma(G)$. Find the random variable $Y$ explicitly in this case.

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• Paper 2, Section II, J

  Probability and Measure

  Part II, 2009

  State Kolmogorov's zero-one law.

  State Birkhoff's almost everywhere ergodic theorem and von Neumann's $L^{p}$-ergodic theorem.

  State the strong law of large numbers for independent and identically distributed integrable random variables, and use the results above to prove it.

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• Paper 1, Section II, J

  Probability and Measure

  Part II, 2009

  Let $(E, \mathcal{E}, \mu)$ be a measure space. Explain what is meant by a simple function on $(E, \mathcal{E}, \mu)$ and state the definition of the integral of a simple function with respect to $\mu$.

  Explain what is meant by an integrable function on $(E, \mathcal{E}, \mu)$ and explain how the integral of such a function is defined.

  State the monotone convergence theorem.

  Show that the following map is linear

  $$f \mapsto \mu(f): L^{1}(E, \mathcal{E}, \mu) \rightarrow \mathbb{R}$$

  where $\mu(f)$ denotes the integral of $f$ with respect to $\mu$.

  [You may assume without proof any fact concerning simple functions and their integrals. You are not expected to prove the monotone convergence theorem.]

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• Paper 1, Section II, I

  Probability and Measure

  Part II, 2010

  State Carathéodory's extension theorem. Define all terms used in the statement.

  Let $\mathcal{A}$ be the ring of finite unions of disjoint bounded intervals of the form

  $$A=\bigcup_{i=1}^{m}\left(a_{i}, b_{i}\right]$$

  where $m \in \mathbb{Z}^{+}$and $a_{1}<b_{1}<\ldots<a_{m}<b_{m}$. Consider the set function $\mu$ defined on $\mathcal{A}$ by

  $$\mu(A)=\sum_{i=1}^{m}\left(b_{i}-a_{i}\right)$$

  You may assume that $\mu$ is additive. Show that for any decreasing sequence $\left(B_{n}: n \in \mathbb{N}\right)$ in $\mathcal{A}$ with empty intersection we have $\mu\left(B_{n}\right) \rightarrow 0$ as $n \rightarrow \infty$.

  Explain how this fact can be used in conjunction with Carathéodory's extension theorem to prove the existence of Lebesgue measure.

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• Paper 2, Section II, I

  Probability and Measure

  Part II, 2010

  Show that any two probability measures which agree on a $\pi$-system also agree on the $\sigma$-algebra generated by that $\pi$-system.

  State Fubini's theorem for non-negative measurable functions.

  Let $\mu$ denote Lebesgue measure on $\mathbb{R}^{2}$. Fix $s \in[0,1)$. Set $c=\sqrt{1-s^{2}}$ and $\lambda=\sqrt{c}$. Consider the linear maps $f, g, h: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

  $$f(x, y)=\left(\lambda^{-1} x, \lambda y\right), \quad g(x, y)=(x, s x+y), \quad h(x, y)=(x-s y, y)$$

  Show that $\mu=\mu \circ f^{-1}$ and that $\mu=\mu \circ g^{-1}$. You must justify any assertion you make concerning the values taken by $\mu$.

  Compute $r=f \circ h \circ g \circ f$. Deduce that $\mu$ is invariant under rotations.

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• Paper 3, Section II, I

  Probability and Measure

  Part II, 2010

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables with common density function

  $$f(x)=\frac{1}{\pi\left(1+x^{2}\right)}$$

  Fix $\alpha \in[0,1]$ and set

  $$Y_{n}=\operatorname{sgn}\left(X_{n}\right)\left|X_{n}\right|^{\alpha}, \quad S_{n}=Y_{1}+\ldots+Y_{n}$$

  Show that for all $\alpha \in[0,1]$ the sequence of random variables $S_{n} / n$ converges in distribution and determine the limit.

  [Hint: In the case $\alpha=1$ it may be useful to prove that $\mathbb{E}\left(e^{i u X_{1}}\right)=e^{-|u|}$, for all $\left.u \in \mathbb{R} .\right]$

  Show further that for all $\alpha \in[0,1 / 2)$ the sequence of random variables $S_{n} / \sqrt{n}$ converges in distribution and determine the limit.

  [You should state clearly any result about random variables from the course to which you appeal. You are not expected to evaluate explicitly the integral

  $$m(\alpha)=\int_{0}^{\infty} \frac{x^{\alpha}}{\pi\left(1+x^{2}\right)} d x$$

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• Paper 4, Section II, I

  Probability and Measure

  Part II, 2010

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent normal random variables having mean 0 and variance 1 . Set $S_{n}=X_{1}+\ldots+X_{n}$ and $U_{n}=S_{n}-\left\lfloor S_{n}\right\rfloor$. Thus $U_{n}$ is the fractional part of $S_{n}$. Show that $U_{n}$ converges to $U$ in distribution, as $n \rightarrow \infty$ where $U$ is uniformly distributed on $[0,1]$.

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• Paper 1, Section II, $26 \mathrm{~K}$

  Probability and Measure

  Part II, 2011

  (i) Let $(E, \mathcal{E}, \mu)$ be a measure space and let $1 \leqslant p<\infty$. For a measurable function $f$, let $\|f\|_{p}=\left(\int|f|^{p} d \mu\right)^{1 / p}$. Give the definition of the space $L^{p}$. Prove that $\left(L^{p},\|\cdot\|_{p}\right)$ forms a Banach space.

  [You may assume that $L^{p}$ is a normed vector space. You may also use in your proof any other result from the course provided that it is clearly stated.]

  (ii) Show that convergence in probability implies convergence in distribution.

  [Hint: Show the pointwise convergence of the characteristic function, using without proof the inequality $\left|e^{i y}-e^{i x}\right| \leqslant|x-y|$ for $x, y \in \mathbb{R}$.]

  (iii) Let $\left(\alpha_{j}\right)_{j \geqslant 1}$ be a given real-valued sequence such that $\sum_{j=1}^{\infty} \alpha_{j}^{2}=\sigma^{2}<\infty$. Let $\left(X_{j}\right)_{j \geqslant 1}$ be a sequence of independent standard Gaussian random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let

  $$Y_{n}=\sum_{j=1}^{n} \alpha_{j} X_{j}$$

  Prove that there exists a random variable $Y$ such that $Y_{n} \rightarrow Y$ in $L^{2}$.

  (iv) Specify the distribution of the random variable $Y$ defined in part (iii), justifying carefully your answer.

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• Paper 2, Section II, $26 \mathrm{~K}$

  Probability and Measure

  Part II, 2011

  (i) Define the notions of a $\pi$-system and a $d$-system. State and prove Dynkin's lemma.

  (ii) Let $\left(E_{1}, \mathcal{E}_{1}, \mu_{1}\right)$ and $\left(E_{2}, \mathcal{E}_{2}, \mu_{2}\right)$ denote two finite measure spaces. Define the $\sigma$ algebra $\mathcal{E}_{1} \otimes \mathcal{E}_{2}$ and the product measure $\mu_{1} \otimes \mu_{2}$. [You do not need to verify that such a measure exists.] State (without proof) Fubini's Theorem.

  (iii) Let $(E, \mathcal{E}, \mu)$ be a measure space, and let $f$ be a non-negative Borel-measurable function. Let $G$ be the subset of $E \times \mathbb{R}$ defined by

  $$G=\{(x, y) \in E \times \mathbb{R}: 0 \leqslant y \leqslant f(x)\}$$

  Show that $G \in \mathcal{E} \otimes \mathcal{B}(\mathbb{R})$, where $\mathcal{B}(\mathbb{R})$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$. Show further that

  $$\int f d \mu=(\mu \otimes \lambda)(G)$$

  where $\lambda$ is Lebesgue measure.

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• Paper 3, Section II, 25K

  Probability and Measure

  Part II, 2011

  (i) State and prove Kolmogorov's zero-one law.

  (ii) Let $(E, \mathcal{E}, \mu)$ be a finite measure space and suppose that $\left(B_{n}\right)_{n \geqslant 1}$ is a sequence of events such that $B_{n+1} \subset B_{n}$ for all $n \geqslant 1$. Show carefully that $\mu\left(B_{n}\right) \rightarrow \mu(B)$, where $B=\cap_{n=1}^{\infty} B_{n}$.

  (iii) Let $\left(X_{i}\right)_{i \geqslant 1}$ be a sequence of independent and identically distributed random variables such that $\mathbb{E}\left(X_{1}^{2}\right)=\sigma^{2}<\infty$ and $\mathbb{E}\left(X_{1}\right)=0$. Let $K>0$ and consider the event $A_{n}$ defined by

  $$A_{n}=\left\{\frac{S_{n}}{\sqrt{n}} \geqslant K\right\}, \quad \text { where } \quad S_{n}=\sum_{i=1}^{n} X_{i}$$

  Prove that there exists $c>0$ such that for all $n$ large enough, $\mathbb{P}\left(A_{n}\right) \geqslant c$. Any result used in the proof must be stated clearly.

  (iv) Prove using the results above that $A_{n}$ occurs infinitely often, almost surely. Deduce that

  $$\limsup _{n \rightarrow \infty} \frac{S_{n}}{\sqrt{n}}=\infty$$

  almost surely.

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• Paper 4, Section II, K

  Probability and Measure

  Part II, 2011

  (i) State and prove Fatou's lemma. State and prove Lebesgue's dominated convergence theorem. [You may assume the monotone convergence theorem.]

  In the rest of the question, let $f_{n}$ be a sequence of integrable functions on some measure space $(E, \mathcal{E}, \mu)$, and assume that $f_{n} \rightarrow f$ almost everywhere, where $f$ is a given integrable function. We also assume that $\int\left|f_{n}\right| d \mu \rightarrow \int|f| d \mu$ as $n \rightarrow \infty$.

  (ii) Show that $\int f_{n}^{+} d \mu \rightarrow \int f^{+} d \mu$ and that $\int f_{n}^{-} d \mu \rightarrow \int f^{-} d \mu$, where $\phi^{+}=\max (\phi, 0)$ and $\phi^{-}=\max (-\phi, 0)$ denote the positive and negative parts of a function $\phi$.

  (iii) Here we assume also that $f_{n} \geqslant 0$. Deduce that $\int\left|f-f_{n}\right| d \mu \rightarrow 0$.

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• Paper 4, Section II, J

  Probability and Measure

  Part II, 2012

  State and prove Fatou's lemma. [You may use the monotone convergence theorem.]

  For $(E, \mathcal{E}, \mu)$ a measure space, define $L^{1}:=L^{1}(E, \mathcal{E}, \mu)$ to be the vector space of $\mu$ integrable functions on $E$, where functions equal almost everywhere are identified. Prove that $L^{1}$ is complete for the norm $\|\cdot\|_{1}$,

  $$\|f\|_{1}:=\int_{E}|f| d \mu, \quad f \in L^{1} .$$

  [You may assume that $\|\cdot\|_{1}$ indeed defines a norm on $L^{1}$.] Give an example of a measure space $(E, \mathcal{E}, \mu)$ and of a sequence $f_{n} \in L^{1}$ that converges to $f$ almost everywhere such that $f \notin L^{1}$.

  Now let

  $$\mathcal{D}:=\left\{f \in L^{1}: f \geqslant 0 \text { almost everywhere }, \int_{E} f d \mu=1\right\} \text {. }$$

  If a sequence $f_{n} \in \mathcal{D}$ converges to $f$ in $L^{1}$, does it follow that $f \in \mathcal{D} ?$ If $f_{n} \in \mathcal{D}$ converges to $f$ almost everywhere, does it follow that $f \in \mathcal{D}$ ? Justify your answers.

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• Paper 3, Section II, J

  Probability and Measure

  Part II, 2012

  Carefully state and prove the first and second Borel-Cantelli lemmas.

  Now let $\left(A_{n}: n \in \mathbb{N}\right)$ be a sequence of events that are pairwise independent; that is, $\mathbb{P}\left(A_{n} \cap A_{m}\right)=\mathbb{P}\left(A_{n}\right) \mathbb{P}\left(A_{m}\right)$ whenever $m \neq n$. For $N \geqslant 1$, let $S_{N}=\sum_{n=1}^{N} 1_{A_{n}}$. Show that $\operatorname{Var}\left(S_{N}\right) \leqslant \mathbb{E}\left(S_{N}\right)$.

  Using Chebyshev's inequality or otherwise, deduce that if $\sum_{n=1}^{\infty} \mathbb{P}\left(A_{n}\right)=\infty$, then $\lim _{N \rightarrow \infty} S_{N}=\infty$ almost surely. Conclude that $\mathbb{P}\left(A_{n}\right.$ infinitely often $)=1 .$

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• Paper 2, Section II, J

  Probability and Measure

  Part II, 2012

  The Fourier transform of a Lebesgue integrable function $f \in L^{1}(\mathbb{R})$ is given by

  $$\hat{f}(u)=\int_{\mathbb{R}} f(x) e^{i x u} d \mu(x)$$

  where $\mu$ is Lebesgue measure on the real line. For $f(x)=e^{-a x^{2}}, x \in \mathbb{R}, a>0$, prove that

  $$\hat{f}(u)=\sqrt{\frac{\pi}{a}} e^{-\frac{u^{2}}{4 a}}$$

  [You may use properties of derivatives of Fourier transforms without proof provided they are clearly stated, as well as the fact that $\phi(x)=(2 \pi)^{-1 / 2} e^{-x^{2} / 2}$ is a probability density function.]

  State and prove the almost everywhere Fourier inversion theorem for Lebesgue integrable functions on the real line. [You may use standard results from the course, such as the dominated convergence and Fubini's theorem. You may also use that $g_{t} * f(x):=\int_{\mathbb{R}} g_{t}(x-y) f(y) d y$ where $g_{t}(z)=t^{-1} \phi(z / t), t>0$, converges to $f$ in $L^{1}(\mathbb{R})$ as $t \rightarrow 0$ whenever $\left.f \in L^{1}(\mathbb{R}) .\right]$

  The probability density function of a Gamma distribution with scalar parameters $\lambda>0, \alpha>0$ is given by

  $$f_{\alpha, \lambda}(x)=\lambda e^{-\lambda x}(\lambda x)^{\alpha-1} 1_{[0, \infty)}(x)$$

  Let $0<\alpha<1, \lambda>0$. Is $\widehat{f_{\alpha, \lambda}}$ integrable?

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• Paper 1, Section II, J

  Probability and Measure

  Part II, 2012

  Carefully state and prove Jensen's inequality for a convex function $c: I \rightarrow \mathbb{R}$, where $I \subseteq \mathbb{R}$ is an interval. Assuming that $c$ is strictly convex, give necessary and sufficient conditions for the inequality to be strict.

  Let $\mu$ be a Borel probability measure on $\mathbb{R}$, and suppose $\mu$ has a strictly positive probability density function $f_{0}$ with respect to Lebesgue measure. Let $\mathcal{P}$ be the family of all strictly positive probability density functions $f$ on $\mathbb{R}$ with respect to Lebesgue measure such that $\log \left(f / f_{0}\right) \in L^{1}(\mu)$. Let $X$ be a random variable with distribution $\mu$. Prove that the mapping

  $$f \mapsto \mathbb{E}\left[\log \frac{f}{f_{0}}(X)\right]$$

  has a unique maximiser over $\mathcal{P}$, attained when $f=f_{0}$ almost everywhere.

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• Paper 4, Section II, K

  Probability and Measure

  Part II, 2013

  State Birkhoff's almost-everywhere ergodic theorem.

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables such that

  $$\mathbb{P}\left(X_{n}=0\right)=\mathbb{P}\left(X_{n}=1\right)=1 / 2$$

  Define for $k \in \mathbb{N}$

  $$Y_{k}=\sum_{n=1}^{\infty} X_{k+n-1} / 2^{n}$$

  What is the distribution of $Y_{k} ? \quad$ Show that the random variables $Y_{1}$ and $Y_{2}$ are not independent.

  Set $S_{n}=Y_{1}+\cdots+Y_{n}$. Show that $S_{n} / n$ converges as $n \rightarrow \infty$ almost surely and determine the limit. [You may use without proof any standard theorem provided you state it clearly.]

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• Paper 3, Section II, $25 K$

  Probability and Measure

  Part II, 2013

  Let $X$ be an integrable random variable with $\mathbb{E}(X)=0$. Show that the characteristic function $\phi_{X}$ is differentiable with $\phi_{X}^{\prime}(0)=0$. [You may use without proof standard convergence results for integrals provided you state them clearly.]

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables, all having the same distribution as $X$. Set $S_{n}=X_{1}+\cdots+X_{n}$. Show that $S_{n} / n \rightarrow 0$ in distribution. Deduce that $S_{n} / n \rightarrow 0$ in probability. [You may not use the Strong Law of Large Numbers.]

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• Paper 2, Section II, $26 \mathrm{~K}$

  Probability and Measure

  Part II, 2013

  Let $\left(f_{n}: n \in \mathbb{N}\right)$ be a sequence of non-negative measurable functions defined on a measure space $(E, \mathcal{E}, \mu)$. Show that $\liminf _{n} f_{n}$ is also a non-negative measurable function.

  State the Monotone Convergence Theorem.

  State and prove Fatou's Lemma.

  Let $\left(f_{n}: n \in \mathbb{N}\right)$ be as above. Suppose that $f_{n}(x) \rightarrow f(x)$ as $n \rightarrow \infty$ for all $x \in E$. Show that

  $$\mu\left(\min \left\{f_{n}, f\right\}\right) \rightarrow \mu(f) .$$

  Deduce that, if $f$ is integrable and $\mu\left(f_{n}\right) \rightarrow \mu(f)$, then $f_{n}$ converges to $f$ in $L^{1}$. [Still assume that $f_{n}$ and $f$ are as above.]

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• Paper 1, Section II, $26 \mathrm{~K}$

  Probability and Measure

  Part II, 2013

  State Dynkin's $\pi$-system $/ d$-system lemma.

  Let $\mu$ and $\nu$ be probability measures on a measurable space $(E, \mathcal{E})$. Let $\mathcal{A}$ be a $\pi$-system on $E$ generating $\mathcal{E}$. Suppose that $\mu(A)=\nu(A)$ for all $A \in \mathcal{A}$. Show that $\mu=\nu$.

  What does it mean to say that a sequence of random variables is independent?

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables, all uniformly distributed on $[0,1]$. Let $Y$ be another random variable, independent of $\left(X_{n}: n \in \mathbb{N}\right)$. Define random variables $Z_{n}$ in $[0,1]$ by $Z_{n}=\left(X_{n}+Y\right) \bmod 1$. What is the distribution of $Z_{1}$ ? Justify your answer.

  Show that the sequence of random variables $\left(Z_{n}: n \in \mathbb{N}\right)$ is independent.

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• Paper 4, Section II, K

  Probability and Measure

  Part II, 2014

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent identically distributed random variables. Set $S_{n}=X_{1}+\cdots+X_{n}$.

  (i) State the strong law of large numbers in terms of the random variables $X_{n}$.

  (ii) Assume now that the $X_{n}$ are non-negative and that their expectation is infinite. Let $R \in(0, \infty)$. What does the strong law of large numbers say about the limiting behaviour of $S_{n}^{R} / n$, where $S_{n}^{R}=\left(X_{1} \wedge R\right)+\cdots+\left(X_{n} \wedge R\right)$ ?

  Deduce that $S_{n} / n \rightarrow \infty$ almost surely.

  Show that

  $$\sum_{n=0}^{\infty} \mathbb{P}\left(X_{n} \geqslant n\right)=\infty$$

  Show that $X_{n} \geqslant R n$ infinitely often almost surely.

  (iii) Now drop the assumption that the $X_{n}$ are non-negative but continue to assume that $\mathbb{E}\left(\left|X_{1}\right|\right)=\infty$. Show that, almost surely,

  $$\limsup _{n \rightarrow \infty}\left|S_{n}\right| / n=\infty$$

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• Paper 3, Section II, K

  Probability and Measure

  Part II, 2014

  (i) Let $(E, \mathcal{E}, \mu)$ be a measure space. What does it mean to say that a function $\theta: E \rightarrow E$ is a measure-preserving transformation?

  What does it mean to say that $\theta$ is ergodic?

  State Birkhoff's almost everywhere ergodic theorem.

  (ii) Consider the set $E=(0,1]^{2}$ equipped with its Borel $\sigma$-algebra and Lebesgue measure. Fix an irrational number $a \in(0,1]$ and define $\theta: E \rightarrow E$ by

  $$\theta\left(x_{1}, x_{2}\right)=\left(x_{1}+a, x_{2}+a\right)$$

  where addition in each coordinate is understood to be modulo 1 . Show that $\theta$ is a measurepreserving transformation. Is $\theta$ ergodic? Justify your answer.

  Let $f$ be an integrable function on $E$ and let $\bar{f}$ be the invariant function associated with $f$ by Birkhoff's theorem. Write down a formula for $\bar{f}$ in terms of $f$. [You are not expected to justify this answer.]

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• Paper 2, Section II, $26 K$

  Probability and Measure

  Part II, 2014

  State and prove the monotone convergence theorem.

  Let $\left(E_{1}, \mathcal{E}_{1}, \mu_{1}\right)$ and $\left(E_{2}, \mathcal{E}_{2}, \mu_{2}\right)$ be finite measure spaces. Define the product $\sigma$-algebra $\mathcal{E}=\mathcal{E}_{1} \otimes \mathcal{E}_{2}$ on $E_{1} \times E_{2}$.

  Define the product measure $\mu=\mu_{1} \otimes \mu_{2}$ on $\mathcal{E}$, and show carefully that $\mu$ is countably additive.

  [You may use without proof any standard facts concerning measurability provided these are clearly stated.]

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• Paper 1, Section II, $26 \mathrm{~K}$

  Probability and Measure

  Part II, 2014

  What is meant by the Borel $\sigma$-algebra on the real line $\mathbb{R}$ ?

  Define the Lebesgue measure of a Borel subset of $\mathbb{R}$ using the concept of outer measure.

  Let $\mu$ be the Lebesgue measure on $\mathbb{R}$. Show that, for any Borel set $B$ which is contained in the interval $[0,1]$, and for any $\varepsilon>0$, there exist $n \in \mathbb{N}$ and disjoint intervals $I_{1}, \ldots, I_{n}$ contained in $[0,1]$ such that, for $A=I_{1} \cup \cdots \cup I_{n}$, we have

  $$\mu(A \triangle B) \leqslant \varepsilon$$

  where $A \triangle B$ denotes the symmetric difference $(A \backslash B) \cup(B \backslash A)$.

  Show that there does not exist a Borel set $B$ contained in $[0,1]$ such that, for all intervals $I$ contained in $[0,1]$,

  $$\mu(B \cap I)=\mu(I) / 2$$

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• Paper 4, Section II, J

  Probability and Measure

  Part II, 2015

  (a) State Fatou's lemma.

  (b) Let $X$ be a random variable on $\mathbb{R}^{d}$ and let $\left(X_{k}\right)_{k=1}^{\infty}$ be a sequence of random variables on $\mathbb{R}^{d}$. What does it mean to say that $X_{k} \rightarrow X$ weakly?

  State and prove the Central Limit Theorem for i.i.d. real-valued random variables. [You may use auxiliary theorems proved in the course provided these are clearly stated.]

  (c) Let $X$ be a real-valued random variable with characteristic function $\varphi$. Let $\left(h_{n}\right)_{n=1}^{\infty}$ be a sequence of real numbers with $h_{n} \neq 0$ and $h_{n} \rightarrow 0$. Prove that if we have

  $$\liminf _{n \rightarrow \infty} \frac{2 \varphi(0)-\varphi\left(-h_{n}\right)-\varphi\left(h_{n}\right)}{h_{n}^{2}}<\infty$$

  then $\mathbb{E}\left[X^{2}\right]<\infty$

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• Paper 3 , Section II, J

  Probability and Measure

  Part II, 2015

  (a) Let $(E, \mathcal{E}, \mu)$ be a measure space. What does it mean to say that $T: E \rightarrow E$ is a measure-preserving transformation? What does it mean to say that a set $A \in \mathcal{E}$ is invariant under $T$ ? Show that the class of invariant sets forms a $\sigma$-algebra.

  (b) Take $E$ to be $[0,1)$ with Lebesgue measure on its Borel $\sigma$-algebra. Show that the baker's map $T:[0,1) \rightarrow[0,1)$ defined by

  $$T(x)=2 x-\lfloor 2 x\rfloor$$

  is measure-preserving.

  (c) Describe in detail the construction of the canonical model for sequences of independent random variables having a given distribution $m$.

  Define the Bernoulli shift map and prove it is a measure-preserving ergodic transformation.

  [You may use without proof other results concerning sequences of independent random variables proved in the course, provided you state these clearly.]

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• Paper 2, Section II, J

  Probability and Measure

  Part II, 2015

  (a) Let $(E, \mathcal{E}, \mu)$ be a measure space, and let $1 \leqslant p<\infty$. What does it mean to say that $f$ belongs to $L^{p}(E, \mathcal{E}, \mu)$ ?

  (b) State Hölder's inequality.

  (c) Consider the measure space of the unit interval endowed with Lebesgue measure. Suppose $f \in L^{2}(0,1)$ and let $0<\alpha<1 / 2$.

  (i) Show that for all $x \in \mathbb{R}$,

  $$\int_{0}^{1}|f(y)||x-y|^{-\alpha} d y<\infty$$

  (ii) For $x \in \mathbb{R}$, define

  $$g(x)=\int_{0}^{1} f(y)|x-y|^{-\alpha} d y$$

  Show that for $x \in \mathbb{R}$ fixed, the function $g$ satisfies

  $$|g(x+h)-g(x)| \leqslant\|f\|_{2} \cdot(I(h))^{1 / 2},$$

  where

  $$I(h)=\int_{0}^{1}\left(|x+h-y|^{-\alpha}-|x-y|^{-\alpha}\right)^{2} d y .$$

  (iii) Prove that $g$ is a continuous function. [Hint: You may find it helpful to split the integral defining $I(h)$ into several parts.]

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• Paper 1, Section II, J

  Probability and Measure

  Part II, 2015

  (a) Define the following concepts: a $\pi$-system, a $d$-system and a $\sigma$-algebra.

  (b) State the Dominated Convergence Theorem.

  (c) Does the set function

  $$\mu(A)= \begin{cases}0 & \text { for } A \text { bounded } \\ 1 & \text { for } A \text { unbounded }\end{cases}$$

  furnish an example of a Borel measure?

  (d) Suppose $g:[0,1] \rightarrow[0,1]$ is a measurable function. Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous with $f(0) \leqslant f(1)$. Show that the limit

  $$\lim _{n \rightarrow \infty} \int_{0}^{1} f\left(g(x)^{n}\right) d x$$

  exists and lies in the interval $[f(0), f(1)]$

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• Paper 3, Section II, J

  Probability and Measure

  Part II, 2016

  (a) Define the Borel $\sigma$-algebra $\mathcal{B}$ and the Borel functions.

  (b) Give an example with proof of a set in $[0,1]$ which is not Lebesgue measurable.

  (c) The Cantor set $\mathcal{C}$ is given by

  $$\mathcal{C}=\left\{\sum_{k=1}^{\infty} \frac{a_{k}}{3^{k}}:\left(a_{k}\right) \text { is a sequence with } a_{k} \in\{0,2\} \text { for all } k\right\}$$

  (i) Explain why $\mathcal{C}$ is Lebesgue measurable.

  (ii) Compute the Lebesgue measure of $\mathcal{C}$.

  (iii) Is every subset of $\mathcal{C}$ Lebesgue measurable?

  (iv) Let $f:[0,1] \rightarrow \mathcal{C}$ be the function given by

  $$f(x)=\sum_{k=1}^{\infty} \frac{2 a_{k}}{3^{k}} \quad \text { where } \quad a_{k}=\left\lfloor 2^{k} x\right\rfloor-2\left\lfloor 2^{k-1} x\right\rfloor$$

  Explain why $f$ is a Borel function.

  (v) Using the previous parts, prove the existence of a Lebesgue measurable set which is not Borel.

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• Paper 4, Section II, J

  Probability and Measure

  Part II, 2016

  Give the definitions of the convolution $f * g$ and of the Fourier transform $\widehat{f}$ of $f$, and show that $\widehat{f * g}=\widehat{f} \widehat{g}$. State what it means for Fourier inversion to hold for a function $f$.

  State the Plancherel identity and compute the $L^{2}$ norm of the Fourier transform of the function $f(x)=e^{-x} \mathbf{1}_{[0,1]}$.

  Suppose that $\left(f_{n}\right), f$ are functions in $L^{1}$ such that $f_{n} \rightarrow f$ in $L^{1}$ as $n \rightarrow \infty$. Show that $\widehat{f}_{n} \rightarrow \widehat{f}$ uniformly.

  Give the definition of weak convergence, and state and prove the Central Limit Theorem.

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• Paper 2, Section II, J

  Probability and Measure

  Part II, 2016

  (a) State Jensen's inequality. Give the definition of $\|\cdot\|_{L^{p}}$ and the space $L^{p}$ for $1<p<\infty$. If $\|f-g\|_{L^{p}}=0$, is it true that $f=g$ ? Justify your answer. State and prove Hölder's inequality using Jensen's inequality.

  (b) Suppose that $(E, \mathcal{E}, \mu)$ is a finite measure space. Show that if $1<q<p$ and $f \in L^{p}(E)$ then $f \in L^{q}(E)$. Give the definition of $\|\cdot\|_{L^{\infty}}$ and show that $\|f\|_{L^{p}} \rightarrow\|f\|_{L^{\infty}}$ as $p \rightarrow \infty$.

  (c) Suppose that $1<q<p<\infty$. Show that if $f$ belongs to both $L^{p}(\mathbb{R})$ and $L^{q}(\mathbb{R})$, then $f \in L^{r}(\mathbb{R})$ for any $r \in[q, p]$. If $f \in L^{p}(\mathbb{R})$, must we have $f \in L^{q}(\mathbb{R})$ ? Give a proof or a counterexample.

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• Paper 1, Section II, J

  Probability and Measure

  Part II, 2016

  Throughout this question $(E, \mathcal{E}, \mu)$ is a measure space and $\left(f_{n}\right), f$ are measurable functions.

  (a) Give the definitions of pointwise convergence, pointwise a.e. convergence, and convergence in measure.

  (b) If $f_{n} \rightarrow f$ pointwise a.e., does $f_{n} \rightarrow f$ in measure? Give a proof or a counterexample.

  (c) If $f_{n} \rightarrow f$ in measure, does $f_{n} \rightarrow f$ pointwise a.e.? Give a proof or a counterexample.

  (d) Now suppose that $(E, \mathcal{E})=([0,1], \mathcal{B}([0,1]))$ and that $\mu$ is Lebesgue measure on $[0,1]$. Suppose $\left(f_{n}\right)$ is a sequence of Borel measurable functions on $[0,1]$ which converges pointwise a.e. to $f$.

  (i) For each $n, k$ let $E_{n, k}=\bigcup_{m \geqslant n}\left\{x:\left|f_{m}(x)-f(x)\right|>1 / k\right\}$. Show that $\lim _{n \rightarrow \infty} \mu\left(E_{n, k}\right)=0$ for each $k \in \mathbb{N}$.

  (ii) Show that for every $\epsilon>0$ there exists a set $A$ with $\mu(A)<\epsilon$ so that $f_{n} \rightarrow f$ uniformly on $[0,1] \backslash A$.

  (iii) Does (ii) hold with $[0,1]$ replaced by $\mathbb{R}$ ? Give a proof or a counterexample.

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• Paper 2, Section II, J

  Probability and Measure

  Part II, 2017

  (a) Give the definition of the Fourier transform $\widehat{f}$ of a function $f \in L^{1}\left(\mathbb{R}^{d}\right)$.

  (b) Explain what it means for Fourier inversion to hold.

  (c) Prove that Fourier inversion holds for $g_{t}(x)=(2 \pi t)^{-d / 2} e^{-\|x\|^{2} /(2 t)}$. Show all of the steps in your computation. Deduce that Fourier inversion holds for Gaussian convolutions, i.e. any function of the form $f * g_{t}$ where $t>0$ and $f \in L^{1}\left(\mathbb{R}^{d}\right)$.

  (d) Prove that any function $f$ for which Fourier inversion holds has a bounded, continuous version. In other words, there exists $g$ bounded and continuous such that $f(x)=g(x)$ for a.e. $x \in \mathbb{R}^{d}$.

  (e) Does Fourier inversion hold for $f=\mathbf{1}_{[0,1]}$ ?

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• Paper 3, Section II, J

  Probability and Measure

  Part II, 2017

  (a) Suppose that $\mathcal{X}=\left(X_{n}\right)$ is a sequence of random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Give the definition of what it means for $\mathcal{X}$ to be uniformly integrable.

  (b) State and prove Hölder's inequality.

  (c) Explain what it means for a family of random variables to be $L^{p}$ bounded. Prove that an $L^{p}$ bounded sequence is uniformly integrable provided $p>1$.

  (d) Prove or disprove: every sequence which is $L^{1}$ bounded is uniformly integrable.

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• Paper 4, Section II, J

  Probability and Measure

  Part II, 2017

  (a) Suppose that $(E, \mathcal{E}, \mu)$ is a finite measure space and $\theta: E \rightarrow E$ is a measurable map. Prove that $\mu_{\theta}(A)=\mu\left(\theta^{-1}(A)\right)$ defines a measure on $(E, \mathcal{E})$.

  (b) Suppose that $\mathcal{A}$ is a $\pi$-system which generates $\mathcal{E}$. Using Dynkin's lemma, prove that $\theta$ is measure-preserving if and only if $\mu_{\theta}(A)=\mu(A)$ for all $A \in \mathcal{A}$.

  (c) State Birkhoff's ergodic theorem and the maximal ergodic lemma.

  (d) Consider the case $(E, \mathcal{E}, \mu)=([0,1), \mathcal{B}([0,1)), \mu)$ where $\mu$ is Lebesgue measure on $[0,1)$. Let $\theta:[0,1) \rightarrow[0,1)$ be the following map. If $x=\sum_{n=1}^{\infty} 2^{-n} \omega_{n}$ is the binary expansion of $x$ (where we disallow infinite sequences of $1 \mathrm{~s}$ ), then $\theta(x)=$ $\sum_{n=1}^{\infty} 2^{-n}\left(\omega_{n-1} \mathbf{1}_{n \in E}+\omega_{n+1} \mathbf{1}_{n \in O}\right)$ where $E$ and $O$ are respectively the even and odd elements of $\mathbb{N}$.

  (i) Prove that $\theta$ is measure-preserving. [You may assume that $\theta$ is measurable.]

  (ii) Prove or disprove: $\theta$ is ergodic.

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• Paper 1, Section II, J

  Probability and Measure

  Part II, 2017

  (a) Give the definition of the Borel $\sigma$-algebra on $\mathbb{R}$ and a Borel function $f: E \rightarrow \mathbb{R}$ where $(E, \mathcal{E})$ is a measurable space.

  (b) Suppose that $\left(f_{n}\right)$ is a sequence of Borel functions which converges pointwise to a function $f$. Prove that $f$ is a Borel function.

  (c) Let $R_{n}:[0,1) \rightarrow \mathbb{R}$ be the function which gives the $n$th binary digit of a number in $[0,1$ ) (where we do not allow for the possibility of an infinite sequence of 1 s). Prove that $R_{n}$ is a Borel function.

  (d) Let $f:[0,1)^{2} \rightarrow[0, \infty]$ be the function such that $f(x, y)$ for $x, y \in[0,1)^{2}$ is equal to the number of digits in the binary expansions of $x, y$ which disagree. Prove that $f$ is non-negative measurable.

  (e) Compute the Lebesgue measure of $f^{-1}([0, \infty))$, i.e. the set of pairs of numbers in $[0,1)$ whose binary expansions disagree in a finite number of digits.

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• Paper 4, Section II, J

  Probability and Measure

  Part II, 2018

  Let $(X, \mathcal{A})$ be a measurable space. Let $T: X \rightarrow X$ be a measurable map, and $\mu$ a probability measure on $(X, \mathcal{A})$.

  (a) State the definition of the following properties of the system $(X, \mathcal{A}, \mu, T)$ :

  (i) $\mu$ is T-invariant.

  (ii) $T$ is ergodic with respect to $\mu$.

  (b) State the pointwise ergodic theorem.

  (c) Give an example of a probability measure preserving system $(X, \mathcal{A}, \mu, T)$ in which $\operatorname{Card}\left(T^{-1}\{x\}\right)>1$ for $\mu$-a.e. $x$.

  (d) Assume $X$ is finite and $\mathcal{A}$ is the boolean algebra of all subsets of $X$. Suppose that $\mu$ is a $T$-invariant probability measure on $X$ such that $\mu(\{x\})>0$ for all $x \in X$. Show that $T$ is a bijection.

  (e) Let $X=\mathbb{N}$, the set of positive integers, and $\mathcal{A}$ be the $\sigma$-algebra of all subsets of $X$. Suppose that $\mu$ is a $T$-invariant ergodic probability measure on $X$. Show that there is a finite subset $Y \subseteq X$ with $\mu(Y)=1$.

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• Paper 2, Section II, J

  Probability and Measure

  Part II, 2018

  Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of random variables with $\mathbb{E}\left(\left|X_{n}\right|^{2}\right) \leqslant 1$ for all $n \geqslant 1$.

  (a) Suppose $Z$ is another random variable such that $\mathbb{E}\left(|Z|^{2}\right)<\infty$. Why is $Z X_{n}$ integrable for each $n$ ?

  (b) Assume $\mathbb{E}\left(Z X_{n}\right) \underset{n \rightarrow \infty}{\longrightarrow} 0$ for every random variable $Z$ on $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{E}\left(|Z|^{2}\right)<\infty$. Show that there is a subsequence $Y_{k}:=X_{n_{k}}, k \geqslant 1$, such that

  $$\frac{1}{N} \sum_{k=1}^{N} Y_{k} \underset{N \rightarrow \infty}{\longrightarrow} 0 \text { in } \mathbb{L}^{2}$$

  (c) Assume that $X_{n} \rightarrow X$ in probability. Show that $X \in \mathbb{L}^{2}$. Show that $X_{n} \rightarrow X$ in $\mathbb{L}^{1}$. Must it converge also in $\mathbb{L}^{2} ?$ Justify your answer.

  (d) Assume that the $\left(X_{n}\right)_{n \geqslant 1}$ are independent. Give a necessary and sufficient condition on the sequence $\left(\mathbb{E}\left(X_{n}\right)_{n \geqslant 1}\right)$ for the sequence

  $$Y_{N}=\frac{1}{N} \sum_{k=1}^{N} X_{k}$$

  to converge in $\mathbb{L}^{2}$.

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• Paper 3, Section II, J

  Probability and Measure

  Part II, 2018

  Let $m$ be the Lebesgue measure on the real line. Recall that if $E \subseteq \mathbb{R}$ is a Borel subset, then

  $$m(E)=\inf \left\{\sum_{n \geqslant 1}\left|I_{n}\right|, E \subseteq \bigcup_{n \geqslant 1} I_{n}\right\},$$

  where the infimum is taken over all covers of $E$ by countably many intervals, and $|I|$ denotes the length of an interval $I$.

  (a) State the definition of a Borel subset of $\mathbb{R}$.

  (b) State a definition of a Lebesgue measurable subset of $\mathbb{R}$.

  (c) Explain why the following sets are Borel and compute their Lebesgue measure:

  $$\mathbb{Q}, \quad \mathbb{R} \backslash \mathbb{Q}, \quad \bigcap_{n \geqslant 2}\left[\frac{1}{n}, n\right] .$$

  (d) State the definition of a Borel measurable function $f: \mathbb{R} \rightarrow \mathbb{R}$.

  (e) Let $f$ be a Borel measurable function $f: \mathbb{R} \rightarrow \mathbb{R}$. Is it true that the subset of all $x \in \mathbb{R}$ where $f$ is continuous at $x$ is a Borel subset? Justify your answer.

  (f) Let $E \subseteq[0,1]$ be a Borel subset with $m(E)=1 / 2+\alpha, \alpha>0$. Show that

  $$E-E:=\{x-y: x, y \in E\}$$

  contains the interval $(-2 \alpha, 2 \alpha)$.

  (g) Let $E \subseteq \mathbb{R}$ be a Borel subset such that $m(E)>0$. Show that for every $\varepsilon>0$, there exists $a<b$ in $\mathbb{R}$ such that

  $$m(E \cap(a, b))>(1-\varepsilon) m((a, b)) .$$

  Deduce that $E-E$ contains an open interval around 0 .

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• Paper 1, Section II, J

  Probability and Measure

  Part II, 2018

  (a) Let $X$ be a real random variable with $\mathbb{E}\left(X^{2}\right)<\infty$. Show that the variance of $X$ is equal to $\inf _{a \in \mathbb{R}}\left(\mathbb{E}(X-a)^{2}\right)$.

  (b) Let $f(x)$ be the indicator function of the interval $[-1,1]$ on the real line. Compute the Fourier transform of $f$.

  (c) Show that

  $$\int_{0}^{+\infty}\left(\frac{\sin x}{x}\right)^{2} d x=\frac{\pi}{2}$$

  (d) Let $X$ be a real random variable and $\widehat{\mu_{X}}$ be its characteristic function.

  (i) Assume that $\left|\widehat{\mu_{X}}(u)\right|=1$ for some $u \in \mathbb{R}$. Show that there exists $\theta \in \mathbb{R}$ such that almost surely:

  $$u X \in \theta+2 \pi \mathbb{Z}$$

  (ii) Assume that $\left|\widehat{\mu_{X}}(u)\right|=\left|\widehat{\mu_{X}}(v)\right|=1$ for some real numbers $u$, $v$ not equal to 0 and such that $u / v$ is irrational. Prove that $X$ is almost surely constant. [Hint: You may wish to consider an independent copy of $X$.]

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• Paper 2, Section II, K

  Probability and Measure

  Part II, 2019

  (a) Let $\left(X_{i}, \mathcal{A}_{i}\right)$ for $i=1,2$ be two measurable spaces. Define the product $\sigma$-algebra $\mathcal{A}_{1} \otimes \mathcal{A}_{2}$ on the Cartesian product $X_{1} \times X_{2}$. Given a probability measure $\mu_{i}$ on $\left(X_{i}, \mathcal{A}_{i}\right)$ for each $i=1,2$, define the product measure $\mu_{1} \otimes \mu_{2}$. Assuming the existence of a product measure, explain why it is unique. [You may use standard results from the course if clearly stated.]

  (b) Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which the real random variables $U$ and $V$ are defined. Explain what is meant when one says that $U$ has law $\mu$. On what measurable space is the measure $\mu$ defined? Explain what it means for $U$ and $V$ to be independent random variables.

  (c) Now let $X=\left[-\frac{1}{2}, \frac{1}{2}\right]$, let $\mathcal{A}$ be its Borel $\sigma$-algebra and let $\mu$ be Lebesgue measure. Give an example of a measure $\eta$ on the product $(X \times X, \mathcal{A} \otimes \mathcal{A})$ such that $\eta(X \times A)=\mu(A)=\eta(A \times X)$ for every Borel set $A$, but such that $\eta$ is not Lebesgue measure on $X \times X$.

  (d) Let $\eta$ be as in part (c) and let $I, J \subset X$ be intervals of length $x$ and $y$ respectively. Show that

  $$x+y-1 \leqslant \eta(I \times J) \leqslant \min \{x, y\}$$

  (e) Let $X$ be as in part (c). Fix $d \geqslant 2$ and let $\Pi_{i}$ denote the projection $\Pi_{i}\left(x_{1}, \ldots, x_{d}\right)=\left(x_{1}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{d}\right)$ from $X^{d}$ to $X^{d-1}$. Construct a probability measure $\eta$ on $X^{d}$, such that the image under each $\Pi_{i}$ coincides with the $(d-1)$-dimensional Lebesgue measure, while $\eta$ itself is not the $d$-dimensional Lebesgue measure. $[$ Hint: Consider the following collection of $2 d-1$ independent random variables: $U_{1}, \ldots, U_{d}$ uniformly distributed on $\left[0, \frac{1}{2}\right]$, and $\varepsilon_{1}, \ldots, \varepsilon_{d-1}$ such that $\mathbb{P}\left(\varepsilon_{i}=1\right)=\mathbb{P}\left(\varepsilon_{i}=-1\right)=\frac{1}{2}$ for each $i .]$

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• Paper 3, Section II, K

  Probability and Measure

  Part II, 2019

  (a) Let $X$ and $Y$ be real random variables such that $\mathbb{E}[f(X)]=\mathbb{E}[f(Y)]$ for every compactly supported continuous function $f$. Show that $X$ and $Y$ have the same law.

  (b) Given a real random variable $Z$, let $\varphi_{Z}(s)=\mathbb{E}\left(e^{i s Z}\right)$ be its characteristic function. Prove the identity

  $$\iint g(\varepsilon s) f(x) e^{-i s x} \varphi_{Z}(s) d s d x=\int \hat{g}(t) \mathbb{E}[f(Z-\varepsilon t)] d t$$

  for real $\varepsilon>0$, where is $f$ is continuous and compactly supported, and where $g$ is a Lebesgue integrable function such that $\hat{g}$ is also Lebesgue integrable, where

  $$\hat{g}(t)=\int g(x) e^{i t x} d x$$

  is its Fourier transform. Use the above identity to derive a formula for $\mathbb{E}[f(Z)]$ in terms of $\varphi_{Z}$, and recover the fact that $\varphi_{Z}$ determines the law of $Z$ uniquely.

  (c) Let $X$ and $Y$ be bounded random variables such that $\mathbb{E}\left(X^{n}\right)=\mathbb{E}\left(Y^{n}\right)$ for every positive integer $n$. Show that $X$ and $Y$ have the same law.

  (d) The Laplace transform $\psi_{Z}(s)$ of a non-negative random variable $Z$ is defined by the formula

  $$\psi_{Z}(s)=\mathbb{E}\left(e^{-s Z}\right)$$

  for $s \geqslant 0$. Let $X$ and $Y$ be (possibly unbounded) non-negative random variables such that $\psi_{X}(s)=\psi_{Y}(s)$ for all $s \geqslant 0$. Show that $X$ and $Y$ have the same law.

  (e) Let

  $$f(x ; k)=1_{\{x>0\}} \frac{1}{k !} x^{k} e^{-x}$$

  where $k$ is a non-negative integer and $1_{\{x>0\}}$ is the indicator function of the interval $(0,+\infty)$.

  Given non-negative integers $k_{1}, \ldots, k_{n}$, suppose that the random variables $X_{1}, \ldots, X_{n}$ are independent with $X_{i}$ having density function $f\left(\cdot ; k_{i}\right)$. Find the density of the random variable $X_{1}+\cdots+X_{n}$.

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• Paper 4, Section II, K

  Probability and Measure

  Part II, 2019

  (a) Let $\left(X_{n}\right)_{n \geqslant 1}$ and $X$ be real random variables with finite second moment on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that $X_{n}$ converges to $X$ almost surely. Show that the following assertions are equivalent:

  (i) $X_{n} \rightarrow X$ in $\mathbf{L}^{2}$ as $n \rightarrow \infty$

  (ii) $\mathbb{E}\left(X_{n}^{2}\right) \rightarrow \mathbb{E}\left(X^{2}\right)$ as $n \rightarrow \infty$.

  (b) Suppose now that $\Omega=(0,1), \mathcal{F}$ is the Borel $\sigma$-algebra of $(0,1)$ and $\mathbb{P}$ is Lebesgue measure. Given a Borel probability measure $\mu$ on $\mathbb{R}$ we set

  $$X_{\mu}(\omega)=\inf \left\{x \in \mathbb{R} \mid F_{\mu}(x) \geqslant \omega\right\}$$

  where $F_{\mu}(x):=\mu((-\infty, x])$ is the distribution function of $\mu$ and $\omega \in \Omega$.

  (i) Show that $X_{\mu}$ is a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ with law $\mu$.

  (ii) Let $\left(\mu_{n}\right)_{n \geqslant 1}$ and $\nu$ be Borel probability measures on $\mathbb{R}$ with finite second moments. Show that

  $$\mathbb{E}\left(\left(X_{\mu_{n}}-X_{\nu}\right)^{2}\right) \rightarrow 0 \text { as } n \rightarrow \infty$$

  if and only if $\mu_{n}$ converges weakly to $\nu$ and $\int x^{2} d \mu_{n}(x)$ converges to $\int x^{2} d \nu(x)$ as $n \rightarrow \infty$

  [You may use any theorem proven in lectures as long as it is clearly stated. Furthermore, you may use without proof the fact that $\mu_{n}$ converges weakly to $\nu$ as $n \rightarrow \infty$ if and only if $X_{\mu_{n}}$ converges to $X_{\nu}$ almost surely.]

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• Paper 1, Section II, K

  Probability and Measure

  Part II, 2019

  Let $\mathbf{X}=\left(X_{1}, \ldots, X_{d}\right)$ be an $\mathbb{R}^{d}$-valued random variable. Given $u=\left(u_{1}, \ldots, u_{d}\right) \in$ $\mathbb{R}^{d}$ we let

  $$\phi_{\mathbf{X}}(u)=\mathbb{E}\left(e^{i\langle u, \mathbf{X}\rangle}\right)$$

  be its characteristic function, where $\langle\cdot, \cdot\rangle$ is the usual inner product on $\mathbb{R}^{d}$.

  (a) Suppose $\mathbf{X}$ is a Gaussian vector with mean 0 and covariance matrix $\sigma^{2} I_{d}$, where $\sigma>0$ and $I_{d}$ is the $d \times d$ identity matrix. What is the formula for the characteristic function $\phi_{\mathbf{X}}$ in the case $d=1$ ? Derive from it a formula for $\phi_{\mathbf{X}}$ in the case $d \geqslant 2$.

  (b) We now no longer assume that $\mathbf{X}$ is necessarily a Gaussian vector. Instead we assume that the $X_{i}$ 's are independent random variables and that the random vector $A \mathbf{X}$ has the same law as $\mathbf{X}$ for every orthogonal matrix $A$. Furthermore we assume that $d \geqslant 2$.

  (i) Show that there exists a continuous function $f:[0,+\infty) \rightarrow \mathbb{R}$ such that

  $$\phi_{\mathbf{X}}(u)=f\left(u_{1}^{2}+\ldots+u_{d}^{2}\right)$$

  [You may use the fact that for every two vectors $u, v \in \mathbb{R}^{d}$ such that $\langle u, u\rangle=\langle v, v\rangle$ there is an orthogonal matrix $A$ such that $A u=v$. ]

  (ii) Show that for all $r_{1}, r_{2} \geqslant 0$

  $$f\left(r_{1}+r_{2}\right)=f\left(r_{1}\right) f\left(r_{2}\right) .$$

  (iii) Deduce that $f$ takes values in $(0,1]$, and furthermore that there exists $\alpha \geqslant 0$ such that $f(r)=e^{-r \alpha}$, for all $r \geqslant 0$.

  (iv) What must be the law of $\mathbf{X}$ ?

  [Standard properties of characteristic functions from the course may be used without proof if clearly stated.]

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• Paper 1, Section II, 27K

  Probability and Measure

  Part II, 2020

  (a) Let $(X, \mathcal{F}, \nu)$ be a probability space. State the definition of the space $\mathbb{L}^{2}(X, \mathcal{F}, \nu)$. Show that it is a Hilbert space.

  (b) Give an example of two real random variables $Z_{1}, Z_{2}$ that are not independent and yet have the same law.

  (c) Let $Z_{1}, \ldots, Z_{n}$ be $n$ random variables distributed uniformly on $[0,1]$. Let $\lambda$ be the Lebesgue measure on the interval $[0,1]$, and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Consider the expression

  $$D(f):=\operatorname{Var}\left[\frac{1}{n}\left(f\left(Z_{1}\right)+\ldots+f\left(Z_{n}\right)\right)-\int_{[0,1]} f d \lambda\right]$$

  where Var denotes the variance and $f \in \mathbb{L}^{2}([0,1], \mathcal{B}, \lambda)$.

  Assume that $Z_{1}, \ldots, Z_{n}$ are pairwise independent. Compute $D(f)$ in terms of the variance $\operatorname{Var}(f):=\operatorname{Var}\left(f\left(Z_{1}\right)\right)$.

  (d) Now we no longer assume that $Z_{1}, \ldots, Z_{n}$ are pairwise independent. Show that

  $$\sup D(f) \geqslant \frac{1}{n},$$

  where the supremum ranges over functions $f \in \mathbb{L}^{2}([0,1], \mathcal{B}, \lambda)$ such that $\|f\|_{2}=1$ and $\int_{[0,1]} f d \lambda=0$.

  [Hint: you may wish to compute $D\left(f_{p, q}\right)$ for the family of functions $f_{p, q}=\sqrt{\frac{k}{2}}\left(1_{I_{p}}-1_{I_{q}}\right)$ where $1 \leqslant p, q \leqslant k, I_{j}=\left[\frac{j}{k}, \frac{j+1}{k}\right)$ and $1_{A}$ denotes the indicator function of the subset $\left.A .\right]$

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• Paper 2, Section II, $26 \mathrm{~K}$

  Probability and Measure

  Part II, 2020

  Let $X$ be a set. Recall that a Boolean algebra $\mathcal{B}$ of subsets of $X$ is a family of subsets containing the empty set, which is stable under finite union and under taking complements. As usual, let $\sigma(\mathcal{B})$ be the $\sigma$-algebra generated by $\mathcal{B}$.

  (a) State the definitions of a $\sigma$-algebra, that of a measure on a measurable space, as well as the definition of a probability measure.

  (b) State Carathéodory's extension theorem.

  (c) Let $(X, \mathcal{F}, \mu)$ be a probability measure space. Let $\mathcal{B} \subset \mathcal{F}$ be a Boolean algebra of subsets of $X$. Let $\mathcal{C}$ be the family of all $A \in \mathcal{F}$ with the property that for every $\epsilon>0$, there is $B \in \mathcal{B}$ such that

  $$\mu(A \triangle B)<\epsilon,$$

  where $A \triangle B$ denotes the symmetric difference of $A$ and $B$, i.e., $A \triangle B=(A \cup B) \backslash(A \cap B)$.

  (i) Show that $\sigma(\mathcal{B})$ is contained in $\mathcal{C}$. Show by example that this may fail if $\mu(X)=+\infty$.

  (ii) Now assume that $(X, \mathcal{F}, \mu)=\left([0,1], \mathcal{L}_{[0,1]}, m\right)$, where $\mathcal{L}_{[0,1]}$ is the $\sigma$-algebra of Lebesgue measurable subsets of $[0,1]$ and $m$ is the Lebesgue measure. Let $\mathcal{B}$ be the family of all finite unions of sub-intervals. Is it true that $\mathcal{C}$ is equal to $\mathcal{L}_{[0,1]}$ in this case? Justify your answer.

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• Paper 3, Section II, 26K

  Probability and Measure

  Part II, 2020

  Let $(X, \mathcal{A}, m, T)$ be a probability measure preserving system.

  (a) State what it means for $(X, \mathcal{A}, m, T)$ to be ergodic.

  (b) State Kolmogorov's 0-1 law for a sequence of independent random variables. What does it imply for the canonical model associated with an i.i.d. random process?

  (c) Consider the special case when $X=[0,1], \mathcal{A}$ is the $\sigma$-algebra of Borel subsets, and $T$ is the map defined as

  $$T x=\left\{\begin{array}{l} 2 x, \quad \text { if } x \in\left[0, \frac{1}{2}\right] \\ 2-2 x, \quad \text { if } x \in\left[\frac{1}{2}, 1\right] \end{array}\right.$$

  (i) Check that the Lebesgue measure $m$ on $[0,1]$ is indeed an invariant probability measure for $T$.

  (ii) Let $X_{0}:=1_{\left(0, \frac{1}{2}\right)}$ and $X_{n}:=X_{0} \circ T^{n}$ for $n \geqslant 1$. Show that $\left(X_{n}\right)_{n \geqslant 0}$ forms a sequence of i.i.d. random variables on $(X, \mathcal{A}, m)$, and that the $\sigma$-algebra $\sigma\left(X_{0}, X_{1}, \ldots\right)$ is all of $\mathcal{A}$. [Hint: check first that for any integer $n \geqslant 0, T^{-n}\left(0, \frac{1}{2}\right)$ is a disjoint union of $2^{n}$ intervals of length $1 / 2^{n+1}$.]

  (iii) Is $(X, \mathcal{A}, m, T)$ ergodic? Justify your answer.

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• Paper 4, Section II, K

  Probability and Measure

  Part II, 2020

  (a) State and prove the strong law of large numbers for sequences of i.i.d. random variables with a finite moment of order 4 .

  (b) Let $\left(X_{k}\right)_{k \geqslant 1}$ be a sequence of independent random variables such that

  $$\mathbb{P}\left(X_{k}=1\right)=\mathbb{P}\left(X_{k}=-1\right)=\frac{1}{2}$$

  Let $\left(a_{k}\right)_{k \geqslant 1}$ be a sequence of real numbers such that

  $$\sum_{k \geqslant 1} a_{k}^{2}<\infty$$

  Set

  $$S_{n}:=\sum_{k=1}^{n} a_{k} X_{k}$$

  (i) Show that $S_{n}$ converges in $\mathbb{L}^{2}$ to a random variable $S$ as $n \rightarrow \infty$. Does it converge in $\mathbb{L}^{1}$ ? Does it converge in law?

  (ii) Show that $\|S\|_{4} \leqslant 3^{1 / 4}\|S\|_{2}$.

  (iii) Let $\left(Y_{k}\right)_{k \geqslant 1}$ be a sequence of i.i.d. standard Gaussian random variables, i.e. each $Y_{k}$ is distributed as $\mathcal{N}(0,1)$. Show that then $\sum_{k=1}^{n} a_{k} Y_{k}$ converges in law as $n \rightarrow \infty$ to a random variable and determine the law of the limit.

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• Paper 1, Section II, H

  Probability and Measure

  Part II, 2021

  (a) State and prove Fatou's lemma. [You may use the monotone convergence theorem without proof, provided it is clearly stated.]

  (b) Show that the inequality in Fatou's lemma can be strict.

  (c) Let $\left(X_{n}: n \in \mathbb{N}\right)$ and $X$ be non-negative random variables such that $X_{n} \rightarrow X$ almost surely as $n \rightarrow \infty$. Must we have $\mathbb{E} X \leqslant \sup _{n} \mathbb{E} X_{n}$ ?

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• Paper 2, Section II, H

  Probability and Measure

  Part II, 2021

  Let $(E, \mathcal{E}, \mu)$ be a measure space. A function $f$ is simple if it is of the form $f=\sum_{i=1}^{N} a_{i} 1_{A_{i}}$, where $a_{i} \in \mathbb{R}, N \in \mathbb{N}$ and $A_{i} \in \mathcal{E}$.

  Now let $f:(E, \mathcal{E}, \mu) \rightarrow[0, \infty]$ be a Borel-measurable map. Show that there exists a sequence $f_{n}$ of simple functions such that $f_{n}(x) \rightarrow f(x)$ for all $x \in E$ as $n \rightarrow \infty$.

  Next suppose $f$ is also $\mu$-integrable. Construct a sequence $f_{n}$ of simple $\mu$-integrable functions such that $\int_{E}\left|f_{n}-f\right| d \mu \rightarrow 0$ as $n \rightarrow \infty$.

  Finally, suppose $f$ is also bounded. Show that there exists a sequence $f_{n}$ of simple functions such that $f_{n} \rightarrow f$ uniformly on $E$ as $n \rightarrow \infty$.

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• Paper 3, Section II, $26 \mathrm{H}$

  Probability and Measure

  Part II, 2021

  Show that random variables $X_{1}, \ldots, X_{N}$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ are independent if and only if

  $$\mathbb{E}\left(\prod_{n=1}^{N} f_{n}\left(X_{n}\right)\right)=\prod_{n=1}^{N} \mathbb{E}\left(f_{n}\left(X_{n}\right)\right)$$

  for all bounded measurable functions $f_{n}: \mathbb{R} \rightarrow \mathbb{R}, n=1, \ldots, N$.

  Now let $\left(X_{n}: n \in \mathbb{N}\right)$ be an infinite sequence of independent Gaussian random variables with zero means, $\mathbb{E} X_{n}=0$, and finite variances, $\mathbb{E} X_{n}^{2}=\sigma_{n}^{2}>0$. Show that the series $\sum_{n=1}^{\infty} X_{n}$ converges in $L^{2}(\mathbb{P})$ if and only if $\sum_{n=1}^{\infty} \sigma_{n}^{2}<\infty$.

  [You may use without proof that $\mathbb{E}\left[e^{i u X_{n}}\right]=e^{-u^{2} \sigma_{n}^{2} / 2}$ for $u \in \mathbb{R}$.]

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• Paper 4, Section II, 26H

  Probability and Measure

  Part II, 2021

  Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Show that for any sequence $A_{n} \in \mathcal{F}$ satisfying $\sum_{n=1}^{\infty} \mathbb{P}\left(A_{n}\right)<\infty$ one necessarily has $\mathbb{P}\left(\limsup A_{n} A_{n}\right)=0 .$

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ and $X$ be random variables defined on $(\Omega, \mathcal{F}, \mathbb{P})$. Show that $X_{n} \rightarrow X$ almost surely as $n \rightarrow \infty$ implies that $X_{n} \rightarrow X$ in probability as $n \rightarrow \infty$.

  Show that $X_{n} \rightarrow X$ in probability as $n \rightarrow \infty$ if and only if for every subsequence $X_{n(k)}$ there exists a further subsequence $X_{n(k(r))}$ such that $X_{n(k(r))} \rightarrow X$ almost surely as $r \rightarrow \infty$.

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