A Central Limit Theorem for Multiplicative Systems

Abstract

The central limit theorem was originally proved for independent random variables. The independence is a very strong notion and hard to check. There are various efforts to prove different theorems on independent variables (e.g. strong law of large numbers, central limit theorem, the law of iterated logarithm, convergence theorem of Kolmogorov) under weaker conditions, like mixing, martingale-difference, orthogonality. Among these concepts the weakest one is orthogonality, but this ensures only the validity of law of large numbers.