A General Framework for Self-Normalization

Abstract

In this chapter we provide a general framework for the probability theory of self-normalized processes. We begin by describing another method to prove the large deviation result (3.8) for self-normalized sums of i.i.d. random variables. This approach leads to an exponential family of supermartingales associated with self-normalization in Sect. 10.1. The general framework involves these supermartingales, or weaker variants thereof, called “canonical assumptions” in Sect. 10.2, which also provides a list of lemmas showing a wide range of stochastic models that satisfy these canonical assumptions. Whereas Sect. 9.3 gives exponential inequalities for discrete-time martingales that are related to the canonical assumptions, Sect. 10.3 gives continuous-time analogs of these results.