UMD Probability Qualifying Exams/Jan2010Probability

Problem 1Edit

Let   be a triangular array of Bernoulli random variables with  . Suppose that

 

Find the limiting distribution of  .

SolutionEdit

We will show it converges to a Poisson distribution with parameter  . The characteristic function for the Poisson distribution is  . We show the characteristic function,   converges to  , which implies the result.

 . By our assumptions, this converges to  .

Problem 2Edit

Let   be a sequence of i.i.d. random variables with uniform distribution on  . Prove that

 

exists with probability one and compute its value.

SolutionEdit

Let  .

 .

The random variables   are i.i.d. with finite mean,

 .

Therefore, the strong law of large numbers implies   converges with probability one to  .

So almost surely,   converges to   and   converges to  .

Problem 3Edit

Let   be a square integrable martingale with respect to a nested sequence of  -fields  . Assume  . Prove that

 .

SolutionEdit

Since   is a martingale,   is a non-negative submartingale and   since   is square integrable. Thus   meets the conditions for Doob's Martingale Inequality and the result follows.

Problem 4Edit

The random variable   is defined on a probability space  . Let   and assume   has finite variance. Prove that

 

In words, the dispersion of   about its conditional mean becomes smaller as the  -field grows.


SolutionEdit

   

We will show that the third term vanishes. Then since the second term is nonnegative, the result follows.


  by the law of total probability.

 , since   is  -measurable.

Finally,  

Problem 5Edit

Consider a sequence of random variables   such that  . Assume   and

 

Prove that

(a.)  

(b).  

SolutionEdit

We show  . If   for only finitely many  , then there is a largest index   for which  . We show in contrast that for all  ,  .

First notice,   and  .

Then let   be the event  , then  .

Notice   and  . Therefore   and  . So   and we reach the desired conclusion.

Problem 6Edit

Let   be a nonhomogeneous Poisson process. That is,   a.s.,   has independent increments, and   has a Poisson distribution with parameter

  where   and the rate function   is a continuous positive function.

(a.) Find a continuous strictly increasing function   such that the time-transformed process   is a homogeneous Poisson process with rate parameter 1.

(b.) Let   be the time until the first event in the nonhomogeneous process  . Compute   and  

SolutionEdit