Probability Prelim Exam 1.1 August 2016

Construct an example of a countable family \(\{\mathcal {F}_i\}_{i=1}^{\infty }\) of \(\sigma \)-algebras such that \[ \sigma \left (\bigcup _{i=1}^{\infty }\mathcal {F}_i\right ) \ne \bigcup _{i=1}^{\infty }\mathcal {F}_i. \] Can you do this so that \(\mathcal {F}_i\subset \mathcal {F}_{i+1}\) for all \(i\geq 1\)? Justify your reasoning.

Choose and fix two integers \(N\geq 1\) and \(x\in \{0,1,\ldots ,N\}\). Let \(X_0=x\) and suppose that, conditionally, \(X_{n+1}\) has a \(\operatorname {Binomial}(N,X_n/N)\) distribution given \(X_1,\ldots ,X_n\). More precisely, for all \(n\geq 0\) and \(k=0,\ldots ,N\), \[ \mathbb {P}(X_{n+1}=k\mid \mathcal {F}_n) = \binom {N}{k} \left (\frac {X_n}{N}\right )^k \left (1-\frac {X_n}{N}\right )^{N-k}, \] where \(\mathcal {F}_n=\sigma (\{X_1,\ldots ,X_n\})\) for all \(n\geq 1\). Prove that:

(a) \(X_{\infty }=\lim _{n\to \infty }X_n\) exists a.s. and in \(L^p(\mathbb {P})\) for all \(1\leq p<\infty \).

(b) Prove that \(\mathbb {E}(X_{\infty })=x\) and \(\mathbb {E}(X_{\infty }^2)=Nx\).

(c) Compute the distribution of \(X_{\infty }\).

Let \(n\geq 1\) be a fixed integer and recall that there are \(n!\) permutations of \([n]:=\{1,\ldots ,n\}\). Let \(\sigma :=\{\sigma (1),\ldots ,\sigma (n)\}\) denote an arbitrary permutation of \([n]\), and choose and fix some integer \(j\in [n]\). We say that \(\sigma \) fixes \(j\) if \(\sigma (j)=j\). Now suppose that \(\sigma \) is selected at random, all permutations of \([n]\) equally likely.

(a) What is the probability that \(\sigma \) fixed \(j\)?

(b) What is the expectation of the number of integers in \([n]\) that are fixed by \(\sigma \)?

Construct two uncorrelated random variables that are not independent.

Let \(\{X_n\}_{n=1}^{\infty }\) be i.i.d. random variables, and let \(p>0\) be fixed. Prove that the following are equivalent:

(a) \(\mathbb {E}(|X_1|^p)<\infty \);

(b) \(n^{-1/p}X_n\to 0\) almost surely; and

(c) \(n^{-1/p}\max _{1\leq j\leq n}|X_j|\to 0\) almost surely.

Suppose \(\{X_n\}_{n=1}^{\infty }\) are i.i.d. exponential random variables with mean one; that is, \(\mathbb {P}\{X_n>x\}=e^{-x}\) for all \(x>0\). Prove that \[ \mathbb {P}\left \{\limsup _{n\to \infty }\frac {X_n}{\log n}=1\right \}=1. \]

Suppose \(X_1,X_2,\ldots \) are i.i.d. strictly positive random variables. Compute for all \(n,m,k\geq 1\), \[ \mathbb {E}\left ( \frac {X_1^k+\cdots +X_m^k}{X_1^k+\cdots +X_n^k} \right ). \] You might wish to start by calculating \(\mathbb {E}(X_1^k\mid X_1^k+\cdots +X_n^k)\).

Let \(X\) and \(Y\) be two positive random variables. We say that \(X\) is stochastically dominated by \(Y\) if \(\mathbb {P}\{X>a\}\leq \mathbb {P}\{Y>a\}\) for all \(a>0\). Prove that, in this case, \(\mathbb {E}(X^k)\leq \mathbb {E}(Y^k)\) for all \(k\geq 1\).

Suppose \(X_1,N_1,X_2,N_2,\ldots \) are independent random variables such that the \(X\)’s are Bernoulli\((1/2)\)—that is, \(\mathbb {P}\{X_i=0\}=\mathbb {P}\{X_i=1\}=1/2\) for all \(i\geq 1\)—and each \(N_n\) is Poisson\((n)\); that is, \[ \mathbb {P}\{N_n=k\}=\frac {n^ke^{-n}}{k!} \qquad \text {for all integers }n\geq 1\text { and }k\geq 0. \] Find non-random sequences \(\{a_n\}_{n=1}^{\infty }\) and \(\{b_n\}_{n=1}^{\infty }\) such that, as \(n\to \infty \), \[ \frac {\sum _{i=1}^{N_n}X_i-a_n}{b_n}\Rightarrow \operatorname {N}(0,1). \]

Suppose \(\mathbb {P}\{X_1=0\}=\mathbb {P}\{X_1=1\}=1/2\) and for all integers \(n\geq 1\), \[ \mathbb {P}(X_{n+1}=1\mid \mathcal {F}_n) = 1-\mathbb {P}(X_{n+1}=0\mid \mathcal {F}_n) = \frac {S_n}{n}, \] where \(S_n:=X_1+\cdots +X_n\) and \(\mathcal {F}_n:=\sigma (\{X_1,\ldots ,X_n\})\). Prove that \(n^{-1}S_n=X_1\) almost surely for all \(n\geq 1\).