Fluctuations of Sums

Abstract

In this chapter we consider some basic theory for sums of independent rvs. This includes classical results such as the strong law of large numbers (SLLN) in Section 2.1 and the central limit theorem (CLT) in Section 2.2, but also refinements on these theorems. In Section 2.3 refinements on the CLT are given (asymptotic expansions, large deviations, rates of convergence). Brownian and α-stable motion are introduced in Section 2.4 as weak limits of partial sum processes. They are fundamental stochastic processes which are used throughout this book. This is also the case for the homogeneous Poisson process which occurs as a special renewal counting process in Section 2.5.2. In Sections 2.5.2 and 2.5.3 we study the fluctuations of renewal counting processes and of random sums indexed by a renewal counting process. As we saw in Chapter 1, random sums are of particular interest in insurance; for example, the compound Poisson process is one of the fundamental notions in the field.