Bernoulli Processes

Abstract

In Chapter 5 we investigate Bernoulli processes, where the individual random variables X _t are linear combinations of independent random signs. Random signs are obviously important r.v.s, and occur frequently in connection with “symmetrization procedures”, a very useful tool. Each Bernoulli process is associated with a Gaussian process in a canonical manner, when one replaces the random signs by independent standard Gaussian r.v.s. The Bernoulli process has better tails than the corresponding Gaussian process (it is “subgaussian”) and is bounded whenever the Gaussian process is bounded. There is however a completely different reason for which a Bernoulli process might be bounded, namely that the sum of the absolute values of the coefficients of the random signs remains bounded independently of the index t. A natural question is then to decide whether these two extreme situations are the only fundamental reasons why a Bernoulli process can be bounded, in the sense that a suitable “mixture” of them occurs in every bounded Bernoulli process. This was the “Bernoulli Conjecture”, which has been so brilliantly solved by W. Bednorz and R. Latała. The proof of their fundamental result occupies much of this chapter. Many of the previous ideas it builds upon will be further developed in subsequent chapters.