Probability Prelim Exam 1.1 January 2018

Let \((X_n)\) be a sequence of real-valued random variables and define the tail \(\sigma \)-algebra \(\mathcal {T}=\cap _{n\geq 1}\sigma (X_n,X_{n+1},\ldots )\). Let \(S_n=X_1+\cdots +X_n\). Prove or disprove the following statements.

1. \(\{\lim _{n\to \infty }S_n\text { exists}\}\in \mathcal {T}\).
2. \(\{\overline {\lim }_{n\to \infty }S_n>0\}\in \mathcal {T}\).
3. \(\{\overline {\lim }_{n\to \infty }S_n/c_n>1\}\in \mathcal {T}\) if \(c_n\to \infty \) as \(n\to \infty \).

Let \(\mu \) be a translation invariant \(\sigma \)-finite measure on \((\mathbb {R},\mathcal {B}(\mathbb {R}))\). Assume that there exist \(\alpha <\beta \) such that \(\mu ((\alpha ,\beta ])<\infty \). Prove that there exists a finite positive constant \(c\) such that \(c\mu \) is Lebesgue measure.

Let \((X_n)\) be i.i.d. random variables with mean \(\mu \).

1. Prove that \[ \lim _{n\to \infty } \frac {1}{\log n}\sum _{i=1}^{n}\frac {X_i}{i} = \mu \qquad \text {almost surely.} \]
2. If the random variables also have a finite variance \(\sigma ^2<\infty \), then find \(a_n\) such that \[ a_n\left (\sum _{i=1}^{n}\frac {X_i}{i}-\mu \log n\right ) \] has a weak limit. Identify this weak limit. (You need to prove the weak convergence, once you have identified \(a_n\) and the limiting distribution.)

1. Prove the Borel-Cantelli lemma. That is, prove that if \(\{A_n\}_{n\geq 1}\) is any sequence of events satisfying \(\sum _{n\geq 1}\mathbb {P}(A_n)<\infty \), then \(\mathbb {P}(A_n\text { occurs for infinitely many }n)=0\).
2. Prove that if \((X_n)\) is any sequence of real-valued random variables, then there are constants \(c_n\) so that \(X_n/c_n\to 0\) almost surely as \(n\to \infty \).

Prove or disprove the following statement. If \(X_n\) converges weakly to \(X\) as \(n\to \infty \) and \(Z_n-X_n\) converges weakly to 0 as \(n\to \infty \), then \(Z_n\) converges weakly to \(X\) as \(n\to \infty \).

Let \(X\) and \(Y\) be two standard normal random variables (defined on a common probability space and having positive variances).

1. Prove that they are independent if and only if they are jointly normal (i.e. \((X,Y)\) is a two-dimensional normal random variable) and uncorrelated.
2. Give an example of two standard normal random variables that are uncorrelated but not jointly normal.

Let \((X_n)\) be i.i.d. Cauchy distributed random variables. That is, they have probability density function \(\frac {1}{\pi (1+x^2)}\). Abbreviate \(S_n=X_1+\cdots +X_n\).

1. Prove that \[ \lim _{n\to \infty }\frac {1}{n}\sum _{i=1}^{n}|X_i|=\infty \qquad \text {almost surely.} \]
2. Using the fact that the characteristic function of a Cauchy distributed random variable is given by \(e^{-|t|}\), prove that \(S_n/n\) is again a Cauchy distributed random variable.
3. Deduce that \[ \overline {\lim }_{n\to \infty } S_n/n=\infty \qquad \text {and}\qquad \underline {\lim }_{n\to \infty } S_n/n=-\infty , \qquad \text {almost surely.} \]

Let \((X_n)\) be i.i.d. uniform random variables on \([0,1]\). For each \(n\geq 1\), define \[ Y_n=n\min \{X_k:1\leq k\leq n\}. \] Prove that as \(n\to \infty \), \(Y_n\) converges weakly to an exponential random variable with mean one.

Suppose \((M_n)\) is a martingale with \(M_0=0\) and \(|M_{n+1}-M_n|\leq 1\) for all \(n\geq 0\).

1. Prove that there exists a finite positive constant \(C\) such that \(e^x\leq 1+x+Cx^2\) for all \(x\in [-1,1]\).
2. Prove that \(\mathbb {E}[e^{\lambda M_n}]\leq (1+C\lambda ^2)^n\leq e^{C\lambda ^2n}\) for all \(n\geq 1\) and \(\lambda \in (0,1)\).
3. Prove that \(M_n/n^\delta \to 0\) almost surely for every \(\delta >1/2\).

Give an example of a sequence of random variables \(X_n\) with densities \(f_n(x)\) so that \(X_n\) converges weakly as \(n\to \infty \) to a uniform random variable on \([0,1]\), but \(f_n(x)\) does not converge to 1 for any \(x\in [0,1]\).