UMD Probability Qualifying Exams/Jan2010Probability

Problem 1

Let $\{X_{nk}\},k=1,...,r_n,n=1,2,...$ be a triangular array of Bernoulli random variables with $p_{nk}=P[X_{nk}=1]$. Suppose that $\sum _{k=1}^{r_n}{p}_{nk}\to \lambda \text{ and }\underset{k\le r_n}{\max }{p}_{nk}\to 0.$ Find the limiting distribution of $\sum _{k=1}^{r_n}{X}_{nk}$.

Solution

We will show it converges to a Poisson distribution with parameter $\lambda$. The characteristic function for the Poisson distribution is $e^{\lambda (e^{it}-1)}$. We show the characteristic function, $E[\exp (it\sum _{k=1}^{r_n}{X}_{nk})]$ converges to $e^{\lambda (e^{it}-1)}$, which implies the result.

$\log E[\exp (it\sum _{k=1}^{r_n}{X}_{nk})]=\sum _{k=1}^{r_n}\log ((1-p_{nk})+p_{nk}{e}^{it})=\sum _{k=1}^{r_n}\log (1-p_{nk}(1-e^{it}))=\sum _{k=1}^{r_n}(-p_{nk}(1-e^{it})+O(p_{nk}^2))$. By our assumptions, this converges to $\lambda (e^{it}-1)$.

Problem 2

Let $X_1,X_2,...$ be a sequence of i.i.d. random variables with uniform distribution on $[0,1]$. Prove that $\underset{n\to \mathrm{\infty }}{\lim }(X_1{X}_2\cdots X_n{)}^{1/n}$ exists with probability one and compute its value.

Solution

Let $Y_n=(X_1{X}_2\cdots X_n{)}^{1/n}$.

$\log (Y_n)=\frac{1}{n}\sum _{j=1}^n\log (X_j)$.

The random variables $\log (X_j)$ are i.i.d. with finite mean,

$E[\log (X_j)]=\underset{0}{\overset{1}{\int }}\log (t)dt=-1$.

Therefore, the strong law of large numbers implies $\frac{1}{n}\sum _{j=1}^n\log (X_j)$ converges with probability one to $-1$.

So almost surely, $\log (Y_n)$ converges to $-1$ and $Y_n$ converges to $e^{-1}$.

Problem 3

Let $\{X_n|n=0,1,2,...\}$ be a square integrable martingale with respect to a nested sequence of $\sigma$-fields $\{{ℱ}_n\}$. Assume $E[X_n]=0$. Prove that $P[\underset{1\le k\le n}{\max }|X_k|>ϵ]\le E[X_n^2]/{ϵ}^2$.

Solution

Since $X_n$ is a martingale, $|X_n|$ is a non-negative submartingale and $E[|X_n{|}^2]<\mathrm{\infty }$ since $X_n$ is square integrable. Thus $|X_n|$ meets the conditions for Doob's Martingale Inequality and the result follows.

Problem 4

The random variable $X$ is defined on a probability space $(\mathrm{\Omega },ℱ,P)$. Let ${𝒢}_1\subset {𝒢}_2\subset ℱ$ and assume $X$ has finite variance. Prove that $E[(X-E[X|{𝒢}_2]{)}^2]\le E[(X-E[X|{𝒢}_1]{)}^2].$ In words, the dispersion of $X$ about its conditional mean becomes smaller as the $\sigma$-field grows.

Solution

$E[(X-E[X|{𝒢}_1]{)}^2]=E[((X-E[X|{𝒢}_2])+(E[X|{𝒢}_2]-E[X|{𝒢}_1]){)}^2]$ $=E[(X-E[X|{𝒢}_2]{)}^2]+E[(E[X|{𝒢}_2]-E[X|{𝒢}_1]{)}^2]+2E[(X-E[X|{𝒢}_2])(E[X|{𝒢}_2]-E[X|{𝒢}_1])]$

We will show that the third term vanishes. Then since the second term is nonnegative, the result follows.

$E[(X-E[X|{𝒢}_2])(E[X|{𝒢}_2]-E[X|{𝒢}_1])]=E[E[(X-E[X|{𝒢}_2])(E[X|{𝒢}_2]-E[X|{𝒢}_1])|{𝒢}_2]]$ by the law of total probability.

$E[(X-E[X|{𝒢}_2])(E[X|{𝒢}_2]-E[X|{𝒢}_1])|{𝒢}_2]=(E[X|{𝒢}_2]-E[X|{𝒢}_1])E[(X-E[X|{𝒢}_2])|{𝒢}_2]$, since $(E[X|{𝒢}_2]-E[X|{𝒢}_1])$ is ${𝒢}_2$-measurable.

Finally, $E[(X-E[X|{𝒢}_2])|{𝒢}_2]=E[X|{𝒢}_2]-E[E[X|{𝒢}_2]|{𝒢}_2]=E[X|{𝒢}_2]-E[X|{𝒢}_2]=0$

Problem 5

Consider a sequence of random variables $X_1,X_2,\dots$ such that $X_n=1\text{ or }0$. Assume $P[X_1=1]\ge \alpha$ and $P[X_n=1|X_1,\dots ,X_{n-1}]\ge \alpha >0\text{ for n=2,3,}\dots$ Prove that (a.) $P[X_n=1\text{ for some n}]=1.$ (b). $P[X_n=1\text{ infinitely often}]=1.$

Solution

We show $P[X_n=1\text{ finitely often}]=0.$. If $X_n=1$ for only finitely many $n$, then there is a largest index $T$ for which $X_T=1$. We show in contrast that for all $T$, $P[X_n=0\text{ for all }n\ge T]=0$.

First notice, $P[X_1=0]\le (1-\alpha )$ and $P[X_T=0]=E[P[X_T=0|X_1,X_2,\dots ,X_{T-1}]]\le (1-\alpha )\text{ for T}>1$.

Then let $A_n^{(T)}$ be the event $[X_{T+n-1}=\dots =X_T=0]$, then $P[X_n=0\text{ for all }n\ge T]=P[A_n^{(T)}\text{ occurs for all n}]$.

Notice $P[A_n^{(T)}]=P[X_{T+n-1}=0|A_{n-1}^{(T)}]P[A_{n-1}^{(T)}]\le (1-\alpha )P[A_{n-1}^{(T)}]\text{ for n =2,3,}\dots$ and $P[A_1^{(T)}]=P[X_T=0]\le (1-\alpha )$. Therefore $P[A_n^{(T)}]\le (1-\alpha {)}^n$ and $\underset{n\to \mathrm{\infty }}{\lim }P[A_n^{(T)}]=0$. So $P[X_n=0\text{ for all n}\ge T]=0$ and we reach the desired conclusion.

Problem 6

Let $\{N(t):t\ge 0\}$ be a nonhomogeneous Poisson process. That is, $N(0)=0$ a.s., $N(t)$ has independent increments, and $N(t)-N(s)$ has a Poisson distribution with parameter $\underset{s}{\overset{t}{\int }}\lambda (u)du$ where $0\le s\le t$ and the rate function $\lambda (u)$ is a continuous positive function. (a.) Find a continuous strictly increasing function $h(t)$ such that the time-transformed process $\tilde{N}(t)=N(h(t))$ is a homogeneous Poisson process with rate parameter 1. (b.) Let $T$ be the time until the first event in the nonhomogeneous process $N(t)$. Compute $P[T>t]$ and $P[T>t|N(s)=n]\text{ where }s>t$