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?



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Problem 4Edit



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Problem 5Edit


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Problem 6Edit



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