Transition theorems and almost sure invariance principles for strong martingales.

Abstract

In this thesis we establish almost sure invariance principles (ASIP's) for strong martingales indexed by two parameters. The method we use is that developed by Berkes and Philipp (Ann. Prob., 7, 1979). This thesis is organized in four chapters. In Chapter 1, we give a review of invariance principles. We introduce the origin of the concept of the invariance principle, describe the main methods for proving the ASIP's and state some basic results of the almost sure invariance principle. In Chapter 2, we prove some "transition theorems" which turn two-parameter strong martingales into one-parameter martingales and can help us to prove the ASIP's for two-parameter strong martingales. We also give several simple applications of the transition theorems, such as maximal inequalities with exponential bounds for two-parameter strong martingales and the Prohorov distance between the law of a two-parameter strong martingale and some appropriate normal law. In Chapter 3, we prove our main theorem--the almost sure invariance principle for two-parameter strong martingales and show some applications, including the functional law of the iterated logarithm for two-parameter strong martingales. In Chapter 4--the appendix, we state some known results we want to use and give the proofs of Theorem 2.2.1 in Chapter 2 and Lemma 3.2.10 in Chapter 3.