UMD Probability Qualifying Exams/Aug2008Probability

__TOC__ Problem 1 Let be a Gaussian vector with zero mean and covariance matrix with entries . Find Solution Problem 2 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 . Solution 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 3 Let be independent identically distributed random variables with uniform distribution on [0,1]. For which values of does the series converge almost surely? Solution Problem 4 Solution Problem 5 Solution Problem 6 Solution