Let be a triangular array of Bernoulli random variables with . Suppose that
Find the limiting distribution of .
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 .
Let be a sequence of i.i.d. random variables with uniform distribution on . Prove that
exists with probability one and compute its value.
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 .
Let be a square integrable martingale with respect to a nested sequence of -fields . Assume . Prove that
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.
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.
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.
Consider a sequence of random variables such that . Assume and
We show . If for only finitely many , then there is a largest index for which . We show in contrast that for all , .
Then let be the event , then .
Notice and . Therefore and . So and we reach the desired conclusion.
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