ABSTRACT
We can extend the class of random processes that possess well-defined structure fimctions.13 If f(x) is stationary, then the structure function [equation (52)] will be independent of x. However, it might be that even though f(x) is not stationary [i.e., the autocorrelation function defined in equation (28) does depend on x], it still may happen that the statistics of the differences f(x+t )- f(x) are independent of x. Such a process is called a process with stationary increments.14 As with the definition of stationarity in §9.1.1, a process has stationary increments if and only if all of the moments of f(x+t )- f(x) are independent of x. Even though the autocorrelation function does not exist for this process, the structure function does. Just as, for a stationary process, we define the spectrum to be the Fourier transform of the autocorrelation function [see equation (5)], we can define the spectrum of an SI process in terms of the structure function.
