UMD Probability Qualifying Exams/Aug2008Probability

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Problem 1[edit | edit source]

Let be a Gaussian vector with zero mean and covariance matrix with entries . Find

Solution[edit | edit source]

Problem 2[edit | edit source]

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[edit | edit source]

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[edit | edit source]

Let be independent identically distributed random variables with uniform distribution on [0,1]. For which values of does the series

converge almost surely?

Solution[edit | edit source]

Problem 4[edit | edit source]

Solution[edit | edit source]

Problem 5[edit | edit source]

Solution[edit | edit source]

Problem 6[edit | edit source]

Solution[edit | edit source]