# UMD Probability Qualifying Exams/Aug2008Probability

## Problem 1

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

## Problem 2

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

### Solution

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

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

That is, we want

{\displaystyle {\begin{aligned}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})|{\mathcal {F}}_{n}]\\f(X_{n})&=E[f(X_{n+1})|{\mathcal {F}}_{n}]-g(X_{n})\end{aligned}}}

Therefore, if ${\displaystyle Y_{n}}$  is to be a martingale, we must have

${\displaystyle g(X_{n})=E[f(X_{n+1})|{\mathcal {F}}_{n}]-f(X_{n})}$ .

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

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

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

This explicitly defines the function ${\displaystyle g}$  and verifies that ${\displaystyle Y_{n}}$  is a martingale.

## Problem 3

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