Probability Prelim Exam 1.1 August 2018

Let \(\mu \) be a translation invariant \(\sigma \)-additive measure on \((\mathbb {R},\mathcal {B}(\mathbb {R}))\) such that \(\mu ((0,1])<\infty \). Prove that there exists a \(c\in (0,\infty )\) such that \(c\mu \) is Lebesgue measure.

Let \(\{X_i:i\geq 1\}\) be a sequence of non-negative, i.i.d. random variables such \(\mathbb {E}[X_1]=\infty \). Prove that \[ \lim _{n\to \infty }\frac {X_1+\cdots +X_n}{n}=\infty \qquad \text {almost surely.} \]

Let \(\mu \) and \(\nu \) be probability measures on \((\mathbb {R},\mathcal {B}(\mathbb {R}))\). Let \(\lambda =\frac {1}{2}\mu +\frac {1}{2}\nu \). Let \(\mathcal {F}\) be a \(\sigma \)-algebra contained in \(\mathcal {B}(\mathbb {R})\). Let \(\mu _{\mathcal {F}}\), \(\nu _{\mathcal {F}}\), and \(\lambda _{\mathcal {F}}\) be the restrictions of \(\mu \), \(\nu \), and \(\lambda \) to \(\mathcal {F}\), respectively.

1. Prove that \(\mu _{\mathcal {F}}\ll \lambda _{\mathcal {F}}\) and \(\nu _{\mathcal {F}}\ll \lambda _{\mathcal {F}}\).
2. What is \(f_\mu =\frac {d\mu _{\mathcal {F}}}{d\lambda _{\mathcal {F}}}\) in terms of \(\frac {d\mu }{d\lambda }\)? Of course, a similar relationship holds for \(f_\nu =\frac {d\nu _{\mathcal {F}}}{d\lambda _{\mathcal {F}}}\).
3. For a bounded \(\mathcal {B}(\mathbb {R})\)-measurable function \(g\), compute \(\mathbb {E}^\lambda [g\mid \mathcal {F}]\) in terms of \(\mathbb {E}^\mu [g\mid \mathcal {F}]\), \(\mathbb {E}^\nu [g\mid \mathcal {F}]\), \(f_\mu \), and \(f_\nu \)?

Let \(\{X_i:i\geq 1\}\) be i.i.d. integrable random variables. Prove that \[ \lim _{n\to \infty } \frac {1}{\log n}\sum _{i=1}^{n}\frac {X_i}{i} = \mathbb {E}[X_1]. \]

Let \(\{X_i:i\geq 1\}\) be i.i.d. random variables distributed uniformly on \([0,1]\). Prove that the distribution of \[ \frac {4\sum _{i=1}^{n} iX_i-n^2}{n^{3/2}} \] converges weakly and identify the limiting distribution.

Let \(\{X_n:n\geq 1\}\) be a stochastic process adapted to a filtration \(\{\mathcal {F}_n:n\geq 1\}\) and satisfying \(\mathbb {E}[|X_n|]<\infty \) for all \(n\geq 1\). Prove that \(\{X_n:n\geq 1\}\) is a martingale if and only if \(\mathbb {E}[X_T]=\mathbb {E}[X_1]\) for all bounded stopping times \(T\).

Let \(\{X_n:n\geq 1\}\) be a martingale with respect to filtration \(\{\mathcal {F}_n:n\geq 1\}\), and let \(T\) be a stopping time with respect to \(\{\mathcal {F}_n:n\geq 1\}\). Prove the following two forms of the optional stopping theorem:

1. If \(T\) is almost surely bounded then \(\mathbb {E}[X_T]=\mathbb {E}[X_1]\),
2. If \(\mathbb {E}[\max _{1\leq n\leq T}|X_n|]<\infty \), then \(\mathbb {E}[X_T]=\mathbb {E}[X_1]\).

Prove the identity \[ \frac {\sin \theta }{\theta } = \int _0^1 e^{i\theta (2x-1)}\,dx. \] Then use this to show that \[ \frac {\sin \theta }{\theta } = \prod _{n=1}^{\infty }\cos (\theta /2^n). \] Hint: Use that a Uniform\((0,1)\) random variable can be written as an infinite weighted sum of iid Bernoulli\((1/2)\) random variables.

Let \(Z_1,Z_2,\ldots \) be iid \(N(0,1)\) random variables. First prove Mills’ ratio \[ \mathbb {P}(Z_i>\lambda ) \leq \frac {1}{\lambda \sqrt {2\pi }}e^{-\lambda ^2/2} \] for any \(\lambda >0\). Use this to show that for any \(\epsilon >0\) \[ \mathbb {P}\left ( \limsup _{n\to \infty } \frac {\max _{k\leq n}Z_k}{\sqrt {(2-\epsilon )\log n}} >1 \right )=1. \]

Show that for every \(\rho \in (-1,1)\) the function \[ f(x,y) = \frac {1}{2\pi \sqrt {1-\rho ^2}} \exp \left ( -\frac {x^2-2\rho xy+y^2}{2(1-\rho ^2)} \right ) \] is a pdf on \(\mathbb {R}^2\), and that if \((X,Y)\) is a pair of random variables with pdf \(f\) then \[ \mathbb {P}(XY<0)=\frac {1}{\pi }\arccos \rho . \]