Random Fields

Abstract

We have used the term stochastic process to denote a collection of random variables indexed by a single real parameter. In other words, the parameter space is a subset of the real line and usually an interval. In most applications, this parameter is interpreted as time. There are many applications where it is more appropriate to consider collections of random variables indexed by points in a more general parameter space. For example, in problems involving propagation of electromagnetic waves through random media, the natural parameter space is a subset of R⁴, representing space and time. A similar example is the velocity field in turbulence theory. The term random field is often used to denote a collection of random variables with a parameter space which is a subset of Rⁿ. There are other possible parameter spaces. For example, the parameter space can be taken to be a function space of some kind. Such is the case with generalized processes. Alternatively, we can also take the parameter space to be a collection of subsets of Rⁿ. Such, for example, is the situation for random measures, which have already been made use of in connection with second-order stochastic integrals. Generally speaking, the kind of assumptions that we make concerning mutual dependence of the collection of random variables reflects something of the parameter space.