Functional Limit Theorems For Products Of One Stochastic Process

Let {X_n; n≥1} be sequence of random variables, S_n= sum from k=1 to n X_k, n≥1. The study ofpartial sum of {X_n} is very popular in the last Century. Such as the central limit theory, the strong law of large numbers and the law of iterated logarithm. Some scholars hasstudied the asymptotics for products of sums in the recent years. Arnold and Villasenor(1998) proved that (sum from k=1 to n log S_k-n log n+n)/(2n)^1/2(?)N, asn→∞, where N is a standard normal random variable and S_n is the partial sums of exponentialrandom variables whose mean is 1.Rempala and Wesolowski (2002) improved their results, and got the asymptoticdistribution of products of sums of i.i.d, positive square integrable random variables.Qi (2003) and Qi and Lu(2004) extended this result to a general limit theorem coveringthe case when the underling distribution is in the domain of attraction of a stable lawwith index from the interval [1, 2]. Rempala and Wesolowski (2005) extended this resultconsidering {X_{k, i}}_{i=1,…, k}k=1, 2,…to be a triangular array of i.i.d, positive squareintegrable random variables, and Jin and Wang (2006) considered {X_n, n≥1} to be asequence of negatively associated identically distributed random variables.This paper mainly considers about the functional central limit theorem and func-tional law of iterated logarithm of martingales and a kind of stationary ergodic stochas-tic processes related to martingales. We proved the theorems of this paper by the similarmethod in Zhang and Wang's paper. Just as the references mentioned above, we use in-variance principle and the law of iterated logarithm to get the limit distribution. In thefirst chapter, we will introduce the works which some scholars have already obtained.In the second chapter, we will use the invariance principle of martingales to obtain thefunctional limit distribution of products of sums of martingales. At the third chapter, wewill use the invariance principle for the law of iterated logarithm to obtain functionallaw of iterated logarithm and functional central limit theorem for a stationary ergodicstochastic processes related to martingales.