Randomly Stopped Processes with Independent Increments

This chapter is devoted to the study of several general properties of randomly stopped sums of independent random variables and more generally randomly stopped processes with independent increments. Concerning randomly stopped sums of independent random variables, the key result behind the theory of sequential analysis is Wald’s first equation. In this chapter we discuss this result and provide (in some sense) the weakest conditions under which it holds. Concerning randomly stopped processes with independent increments, the work of Burkholder and Gundy dealing with inequalities for the Lp-norm of Brownian motion (Corollary 5.3 of this chapter) has been of paramount importance in the development of our understanding of the behavior of such processes. In particular, their inequalities (and Wald’s equations) can be used to determine the finiteness of expectations of stopping times associated with the processes in question. The aim of this chapter is to provide a decoupling reinterpretation and extensions of these classical inequalities. We will present an extension of these results to all processes continuous on the right with limits from the left, with independent increments and values in general Banach spaces. Our framework permits an extension in a different direction, dealing with inequalities for more general functions (than powers) like exponentials. As an application we present an example connecting boundary crossing by nonrandom functions to boundary crossing by processes with independent increments.