Gaussian Signals, Correlation Matrices, and Sample Path Properties

Abstract

In general, determination of the shape of the sample paths of a random signal X(t) requires knowledge of n-point probabilities P(a ₁ < X(t ₁) < b ₁,. . ., a _a < X(t _n) < b _n) for an arbitrary n, and arbitrary windows a ₁ < b ₁, . . ., a _n < b _n. But usually this information cannot be recovered if the only signal characteristic known is the autocorrelation function. The latter depends on the two-point distributions but does not uniquely determine them. However, in the case of Gaussian signals, the autocorrelations determine not only the two-point probability distributions but also all the n-point probability distributions, so that complete information is available within the second-order theory. For example, this means that you only have to estimate means and covariances to make the model. Also, in the Gaussian universe, weak stationarity implies strict stationarity as defined in Chapter 4. For the sake of simplicity all signals in this chapter are assumed to be real-valued. The chapter ends with a more subtle analysis of sample path properties of stationary signals such as continuity and differentiability; in the Gaussian case the information is particularly complete.