Reducing non-stationary stochastic processes to stationarity by a time deformation (Q1962235)

Let \(Z=(Z(t)\); \(t \in T)\) be a stochastic process in \(L^2(P)\). A necessary and sufficient condition is given (in terms of the correlation function \(r\) of \(Z\)) for the existence and form of a time transformation \(\Phi(t)\) such that under the new time scale the correlation function depends only on the difference of times: \(r(t_1,t_2)=R(\Phi(t_2)-\Phi(t_1))\).