Stationarizing Properties of Random Shifts

Let$\{X(t, \omega), - \infty < t < \infty\}$be a real jointly measurable stochastic process and θ(ω) a real random variable. We define Y(t, ω) = X(t + θ(ω), ω). If θ is independent of X, then for any times {tj, j = 1, ⋯, n} and every Borel measurable g(Y1, ⋯, Yn) with$E \{\|g\lbrack Y(t_1), \cdots, Y(t_n)\rbrack\|\} < \infty$, we find \begin{equation*}\tag{i} E\{g\lbrack Y(t_1), \cdots, Y(t_n)\rbrack \mid \theta(\omega) = \theta_0\} = E \{g\lbrack X(t_1 + \theta_0), \cdots, X(t_n + \theta_0) \rbrack\}\end{equation*} for almost every θ0relative to μ, the measure induced on the real line by θ(ω). When θ is independent of X and uniformly distributed over [ 0, h], then Y is strictly stationary if and only if X is periodically nonstationary in the sense that its joint distributions are invariant under translations of length h. If θ is independent of a periodically nonstationary X and μ is absolutely continuous with respect to Lebesgue measure, then Y is strictly stationary if and only if μ satisfies Beutler's trigonometric moment condition. There are corresponding results in the wide sense case. If X is uniformly continuous in quadratic mean but not necessarily jointly measurable and θ is independent of X(s) and X(t) for every pair s, t, then X(t + θ(ω), ω) is given as a quadratic mean limit of X(t + θn(ω), ω), where the θn(ω) are simple functions converging pointwise to θ(ω). In this case, (i) holds at least for the first two moments of Y, that is, \begin{equation*}\tag{ii}E\{Y(t)\mid \theta(\omega) = \theta_0\} = E\{X(t + \theta_0)\} \text {a.e.} (\mu), E\{Y(s)Y(t)\mid \theta(\omega) = \theta_0\} = E\{X(s + \theta_0)X(t + \theta_0)\} \text {a.e.} (\mu).\end{equation*} If (ii) holds for θ uniformly distributed over [ 0, h], Y is wide sense stationary if and only if X is periodically correlated in the sense that its first two moments are invariant under time translations of length h.