and only needs to be indexed by one variable rather than two variables. Thus, instead of writing, the notation is often abbreviated by the substitution : This also implies that the autocorrelation depends only on , that is The third property says that the second moments must be finite for any time . Motivation The main advantage of wide-sense stationarity is that it places the time-series in the context of Hilbert spaces. Let H be the Hilbert space generated by {x(t)} (that is, the closure of the set of all linear combinations of these random variables in the Hilbert space of all square-integrable random variables on the given probability space). By the positive definiteness of the autocovariance function, it follows from Bochner's theorem that there exists a positive measure on the real line such that H is isomorphic to the Hilbert subspace of L2(μ) generated by {e−2iξ⋅t}. This then gives the following Fourier-type decomposition for a continuous time stationary stochastic process: there exists a stochastic process with orthogonal increments such that, for all where the integral on the right-hand side is interpreted in a suitable (Riemann) sense. The same result holds for a discrete-time stationary process, with the spectral measure now defined on the unit circle. When processing WSS random signals with linear, time-invariant (LTI) filters, it is helpful to think of the correlation function as a linear operator. Since it is a circulant operator (depends only on the difference between the two arguments), its eigenfunctions are the Fourier complex exponentials. Additionally, since the eigenfunctions of LTI operators are also complex exponentials, LTI processing of WSS random signals is highly tractable—all computations can be performed in the frequency domain. Thus, the WSS assumption is widely employed in signal processing algorithms. Definition for complex stochastic process In the case where is a complex stochastic process the autocovariance function is defined as and, in addition to the requirements in , it is required that the pseudo-autocovariance function depends only on the time lag. In formulas, is WSS, if Joint stationarity The concept of stationarity may be extended to two stochastic processes. Joint strict-sense stationarity Two stochastic processes and are called jointly strict-sense stationary if their joint cumulative distribution remains unchanged under time shifts, i.e. if Joint (M + N)th-order stationarity Two random processes and is said to be jointly (M + N)-th-order stationary if: Joint weak or wide-sense stationarity Two stochastic processes