Comparison principles for Gaussian processes

Abstract

Here we demonstrate how the study of Gaussian stochastic processes is possible by comparing their covariance functions. For this purpose, we need some preordering relations in the class Φ₊(T) of real-valued positive functions given on a set T of parameters. As a rule, T will be a separable metric space. In particular, the following situations are of especial interest to us:

• T = {1, 2,…, m} where m ≥ 1 is a natural number;

• T = N;

• T = [0,1];

• T = R or T = R ₊ = {t ∈ R : t ≥ 0}.