Our aim in this final chapter is to indicate how basic distributional theory for stochastic processes, alias random functions, may be developed from the considerations of Chapters 7 and 9. This is primarily for reference and for readers with a potential interest in the topic. The theory will be first illustrated by a discussion of the definition of the Wiener process, and conditions for sample function continuity. This will be complemented, and the chapter completed with a sketch of construction and basic properties of point processes and random measures in a purely measure-theoretic framework, consistent with the nontopological flavor of the entire volume.
Random functions and stochastic processes
In this section we introduce some basic distributional theory for stochastic processes and random functions, using the product space measures of Chapter 7 and the random element concepts of Chapter 9.
By a stochastic process one traditionally means a family of real random variables {ξt: t ∈ T} (ξ, = ξt(ω)) on a probability space (Ω, F, P), T being a set indexing the ξt. If T = {1,2,3,…} or {…, −2, −1,0,1,2,…} the family {ξn: n =1,2,…} or {ξn: n = …,−2, −1,0,1,2,∧…} is referred to as a stochastic sequence or discrete parameter stochastic process, whereas {ξt: t ∈ T} is termed a continuous parameter stochastic process if T is an interval (finite or infinite).