Preliminary Examination 2000: Probability

Let \(X_1,X_2,X_3,\ldots \) be independent and exponential with mean 1. Use the Borel–Cantelli lemmas to show that \(\limsup _{n\to \infty }X_n/\log n=1\) a.s.

Let \(X\) be such that \(\mathbb {E}[X^+]=\infty \) and \(\mathbb {E}[X^-]<\infty \), and let \(X_1,X_2,X_3,\ldots \) be i.i.d. with common distribution that of \(X\). With \(S_n=X_1+X_2+\cdots +X_n\), show that \(\lim _{n\to \infty }S_n/n=\infty \) a.s.

Let \(X_1,X_2,X_3,\ldots \) be i.i.d. nonnegative random variables with mean 1 and variance 1, and put \(S_n=X_1+X_2+\cdots +X_n\). Show that \(2(\sqrt {S_n}-\sqrt {n})\) converges in distribution to a standard normal as \(n\to \infty \).

Let \(X_1,X_2,X_3,\ldots \) be a sequence of random variables, each of which has all moments finite, and suppose that \(\lim _{n\to \infty }\mathbb {E}[(X_n)^k]=k!\) for each \(k\ge 1\). Show that \(X_n\) converges in distribution.

Let \(X_1,X_2,X_3,\ldots \) be i.i.d. with finite mean and variance, and put \(S_n=X_1+X_2+\cdots +X_n\). Show that \(S_n-n\mathbb {E}[X_1]\) and \((S_n-n\mathbb {E}[X_1])^2-n\operatorname {Var}(X_1)\) are martingales.

Let \(\mathbb {P}\{X=1\}=\mathbb {P}\{X=-1\}=1/2\), let \(X_1,X_2,X_3,\ldots \) be i.i.d. with common distribution that of \(X\), and put \(S_n=X_1+X_2+\cdots +X_n\). Define \(T=\min \{n\ge 0:S_n=-A\text { or }S_n=B\}\), where \(A\) and \(B\) are positive integers. Using the martingales of problem 5, find \(\mathbb {P}\{S_T=B\}\) and \(\mathbb {E}[T]\).

Let \(P\) be the transition matrix for a finite irreducible aperiodic Markov chain, and assume that \(P\) is doubly stochastic (row and column sums are 1). Find the limit \(\lim _{n\to \infty }P_{ij}^n\) for all \(i\) and \(j\), and provide justification.

Consider the Markov chain in the set of nonnegative integers with transitions \(P(n,0)=1-P(n,n+1)=p_n>0\) for each \(n\ge 0\). Give necessary and sufficient conditions on the sequence \(\{p_n\}\) for the chain to be (a) positive recurrent, (b) null recurrent, and (c) transient.