ABSTRACT
Stochastic processes have been treated so far mainly in connection with martingales, although a general definition was given: a stochastic process is a function X of two variables t and ω, t ∈ T, ω ∈ Ω, where (Ω, R, P) is a probability space and for each t, X(t, •) is measurable on Ω. Taking T to be the set of positive integers, any sequence of random variables is a stochastic process. In much of the more classical theory of processes, T is a subset of the real line. But by the 1950s, if not before, it began to be realized that there are highly irregular random processes, useful in representing or approximating “noise,” for example, which are in a sense defined over the line, but which do not have values at points t. Instead, “integrals” W(ƒ) = ∫W(t)f(t) dt are defined only if ƒ has some smoothness and/or other regularity properties. Thus an appropriate index set T for the process may be a set of functions on ℝ rather than a subset of ℝ. Such processes are also useful where we may have random functions not only changing in time but defined also on space, so that T may be a set of smooth functions of space as well as, or instead of, time variables. At any rate, the beginnings of the theory of stochastic processes, and a basic existence theorem, hold for an arbitrary index set T without any structure. Then what is from many points of view the single most important process, the “Wiener process” or “Brownian motion,” will be defined and studied. If Xi are i.i.d. variables with mean 0 and finite variance σ 2, and Sn = X 1 + ⋯ + Xn , lim sup n →∞ Sn /(2n log log n)1/2 = 1, (the law of the iterated logarithm), will be proved in §12.5, using Brownian motion.
