Distributional Limit Theorems for Partial Sums

Abstract

In this chapter we study convergence in distribution in settings involving sequences (S _n : n = 1, 2,...), where for each n, S _n = X ₁+... + X _n is the n ^th partial sum of a series of independent random variables. Our first result is that convergence in distribution of (S _n ) is equivalent to a.s. convergence. Thereafter, we specialize to the case in which (X ₁, X ₂,...) is an iid sequence. Further limit theorems involving more general sums of independent random variables will be found in Chapter 16.