Stationary Processes

Abstract

Consider a random process Xt(ω) indexed by a parameter t∈T for which an additive group operation is defined. Typical examples are those in which T is the set of real numbers or the set of lattice points t = 0, ±1,… on the real line. More generally T might be all the points in k-dimensional Euclidian space or the lattice points in such a space. We shall say that the process is strictly stationary if the random variables

$${{X}_{{{{t}_{1}}}}},{{X}_{{{{t}_{2}}}}}, \ldots ,{{X}_{{{{t}_{m}}}}}$$

((1))

have the same joint probability distribution as the random variables

$${{X}_{{{{t}_{1}} + h}}}, \ldots ,{{X}_{{{{t}_{m}} + h}}}$$

((2))

for any positive integer m, any t ₁,..., t _m and all h in T. This can be phrased more succinctly by stating that the probability structure of the process is invariant under parameter translation. The finite dimensional joint probability distributions therefore depend on t ₁,..., t _m only through the differences t ₂ — t ₁,..., t _m — t ₁. It is worthwhile giving a few examples of stationary processes as they arise as models of various types of natural phenomena. The examples we give are set in an engineering or physics context. This is natural since it is here that stationary processes have been thought of as natural models of natural phenomena most frequently.