UMD Probability Qualifying Exams/Aug2008Probability
Problem 1Edit
Let be a Gaussian vector with zero mean and covariance matrix with entries . Find 
SolutionEdit
Problem 2Edit
Let be a Markov chain on the state space having transition matrix with elements . Let be the function with and . Find a function such that
is a martingale relative to the filtration generated by the process . 
SolutionEdit
Notice that since are measurable functions, then is composed of linear combinations of measurable functions and hence is adapted. Furthermore, for any , is finite everywhere, hence is .
Therefore, we only need to check the conditional martingale property, i.e. we want to show .
That is, we want
Therefore, if is to be a martingale, we must have
.
Since , we can compute the right hand side without too much work.
This explicitly defines the function and verifies that is a martingale.
Problem 3Edit
Let be independent identically distributed random variables with uniform distribution on [0,1]. For which values of does the series
converge almost surely? 
SolutionEdit
Problem 4Edit

SolutionEdit
Problem 5Edit

SolutionEdit
Problem 6Edit
