Analysis of Linear Stochastic Systems

We shall deal only with processes which evolve at discrete instances of time. Typically, the time index can be k0, k0 + 1, . . . , N , with k0 and N both finite, or it can be the nonnegative integers Z+ = 0, 1, . . ., or it can be all the integers Z. Let T be such an index set, which can be finite or countably infinite. Assume also that there is an underlying probability space (Ω,S, P ) with respect to which all random variables are defined. A discrete time stochastic process is just a family of random variables wk, k ∈ T . This means that for each fixed k, wk is a random variable on the sample space Ω, and that the family wk, k ∈ T is jointly defined on Ω. We also use the notation w(k), k ∈ T to denote the stochastic process. The stochastic process w is thus a function of 2 variables, k denoting the evolution of time, and ω denoting the point in the sample space that the random variables are to be evaluated. If we fix ω, wk(ω) is a function of k only. These are called the sample paths or realizations of the stochastic process. If we take arbitrary, but finite collections of points in T , i1, i2, . . . , in, the family of joint distributions Fwi1 ,wi2 ,··· ,win (w1, w2, · · · , wn) is called the finite dimensional distributions of w. It can be shown that given a family of distributions which satisfy certain consistency properties, there exists a stochastic process w which has the given family of distributions as its finite dimensional distributions.