UMD Probability Qualifying Exams/Jan2010Probability

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Problem 1[edit]

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

Find the limiting distribution of .

Solution[edit]

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 2[edit]

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

exists with probability one and compute its value.

Solution[edit]

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 3[edit]

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

.

Solution[edit]

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 4[edit]

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.


Solution[edit]

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 5[edit]

Consider a sequence of random variables such that . Assume and

Prove that

(a.)

(b).

Solution[edit]

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 6[edit]

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

Solution[edit]