Limit Laws for Generalizations of Martingales

Abstract

Given a sequence of random variables (xk)k ≧ 1 one of the important problems in probability and statistics is the description of the asymptotic properties of the partial sum process S(t) = Σk ≦ t xk. Essentially three types of results are known: strong laws of large numbers (SLLNs), central limit theorems (CLTs) and laws of the iterated logarithm (LILs). All three of them as well as a number of refinements such as the functional versions of the last two or upper and lower class results can be derived from a strong or almost sure approximation of the partial sum process by a Brownian motion (X(t))t ≧ 0. Since as is well known Brownian motion satisfies all laws, the same holds for the partial sum process if the approximation is sharp enough, namely if (1) for some K > O. In other words formula (1) — once it is established — contains all information on the asymptotic behavior of the partial sums in a very compact form. Thus the question is to look for assumptions under which (1) holds.