UMD Probability Qualifying Exams/Jan2010Probability

__TOC__ Problem 1 Let be a triangular array of Bernoulli random variables with . Suppose that Find the limiting distribution of . Solution 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 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 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 Let be a square integrable martingale with respect to a nested sequence of -fields . Assume . Prove that . Solution 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 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 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 Consider a sequence of random variables such that . Assume and Prove that (a.) (b). Solution 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 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