Introduction to Random Fields

Abstract

One means of generalizing denumerable stochastic processes {x _n } with time parameter set ℕ = {0, 1, ... } is to consider random fields {x _t }, where t takes on values in an arbitrary countable parameter set T. Roughly, a random field with denumerable state space S is described by a probability measure μ on the space Ω = S^T of all configurations of values from S on the generalized time set T. In this chapter we discuss certain extensions of Markov chains, called Markov fields which have been important objects of study in the recent development of probability theory. Only some of the highlights of this rich theory will be covered; we concentrate especially on the case T = ℤ = the integers, where the connections with classical Markov chain theory are deepest.