Stationarizing Properties of Random Shifts

Let \ X(t,ω ), - ∞ < t < ∞ \ 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[Y(t₁ ), ⋯ ,Y(tₙ )] |\ < ∞, we find \[ ({i}) E\{ | {g[ Y( t_1 ) , ⋯ ,Y( t_n ) ]} | θ (ω ) = θ _0 \} = E\{ g[ X( t_1 + θ _0 ) , ⋯ ,X( t_n + θ _0 ) ] \} \] for almost every θ ₀ relative 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 + θ ₙ (ω ),ω ), where the θ ₙ (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, ${gathered} ({ii}) E\{ Y(t)| {θ (ω ) = θ _0 } \} = E\{ X(t + θ _0 )\} {a.e..}\,(μ ), \\ E\{ Y(s)Y(t)| {θ (ω ) = } θ _0 \} = E\{ X(s + θ _0 )X(t + θ _0 )\} {a.e.}\,(μ ). \\ {gathered} $ 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.