UMD Probability Qualifying Exams/Aug2008Probability

Problem 1

Let $({\xi }_1,{\xi }_2)$ be a Gaussian vector with zero mean and covariance matrix $D$ with entries $D_{11}=D_{22}=1,D_{12}=D_{21}=1/2$. Find $E({\xi }_1^2{\xi }_2|2{\xi }_1-{\xi }_2)$

Problem 2

Let $X_n,n\ge 0$ be a Markov chain on the state space $\mathrm{X}=\{1,2\}$ having transition matrix $P$ with elements $P_{11}=1/3,P_{12}=2/3,P_{21}=P_{22}=1/2$. Let $f:\mathrm{X}\to ℝ$ be the function with $f(1)=1$ and $f(2)=4$. Find a function $g:\mathrm{X}\to ℝ$ such that $Y_n=f(X_n)-f(X_0)-\sum _{i=0}^{n-1}g(X_i),n\ge 1,$ is a martingale relative to the filtration ${ℱ}_n^X$ generated by the process $X_n$.

Solution

Notice that since $f,g$ are measurable functions, then $Y_n$ is composed of linear combinations of ${ℱ}_n$-measurable functions and hence $Y_n$ is ${ℱ}_n$-adapted. Furthermore, for any $n$, $Y_n$ is finite everywhere, hence is $L^1$.

Therefore, we only need to check the conditional martingale property, i.e. we want to show $Y_n=E(Y_{n+1}|{ℱ}_n$.

That is, we want

$\begin{array}{cc}f(X_n)-f(X_0)-\sum _{i=1}^{n-1}g(X_i)& =E[f(X_{n+1})-f(X_0)-\sum _{i=1}^{n}g(X_i)|{ℱ}_n]\\ f(X_n)& =E[f(X_{n+1})|{ℱ}_n]-g(X_n)\end{array}$

Therefore, if $Y_n$ is to be a martingale, we must have

$g(X_n)=E[f(X_{n+1})|{ℱ}_n]-f(X_n)$.

Since $\mathrm{X}=\{1,2\}$, we can compute the right hand side without too much work.

$g(1)=E[f(X_{n+1})|X_n=1]-f(1)=(1\cdot 1/3+4\cdot 2/3)-1=2$

$g(2)=E[f(X_{n+1})|X_n=2]-f(2)=(1\cdot 1/2+4\cdot 1/2)=-\frac{3}{2}$

This explicitly defines the function $g$ and verifies that $Y_n$ is a martingale.

Problem 3

Let ${\xi }_n$ be independent identically distributed random variables with uniform distribution on [0,1]. For which values of $\alpha >0$ does the series $\sum _{n=1}^{\mathrm{\infty }}({\xi }_n+n^{-\alpha }{)}^{(n^{\alpha +1})}$ converge almost surely?