Qualifying Exam in Analysis

Fall 1998

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1. Let be a sequence of random variables on a probability spzce .
   1. give the precise definition for the following notions of convergence of to a random variable : pointwize a.e.; in probability; in distribution.
   2. show that convergence pointwise a.e. convergence in probability convergence in distribution.
   3. show that for all , pointwise a.e.
   4. show that convergence convergence in probability convergence pointwise for some subsequence .
   
   
   
2. Let be i.i.d. random variables with and . Let , and .
   1. show that for any , converges as to
   2. what can you say about the convergence of the random variables ?
   
   
   
3. Consider the initial value problem
   
   

   where is a smooth function with compact support.

   1. exhibit a solution of the problem in terms of Brownian motion.
   2. exhibit in terms of an explicit integral formula. Verify carefully that the integral formula does satisfy both conditions in the initial-value problem.
   
   
   
4. Let be a bounded open set in , and let be the subspace of consisting of functions which are holomorphic in .
   1. show that for any compact set in , there is a constant so that
      

      for all .

   2. show that is a complete subspace of the space .
   
   
   
5. Define the Sobolev norm of order of a function in by
   

   where is the Fourier transform of . Show that if is any compact set in , and is any sequence of functions supported in which satisfies for some fixed , then there is a subsequence of which converges in .
   
   

6. Let Sobolev norms be defined as in Problem 5. If , show that for any , there is a constant so that the inequality
   

   holds for all .