Stationarizing Properties of Random Shifts

Let $\{ X(t,\omega ), - \infty < t < \infty \} $ be a real jointly measurable stochastic process and $\theta (\omega )$ a real random variable. We define \[ Y(t,\omega ) = X(t + \theta (\omega ),\omega )\]. If $\theta $ is independent of X, then for any times $\{ t_j ,j = 1, \cdots ,n\} $ and every Borel measurable $g(Y_1 , \cdots ,Y_n )$ with $E\{ | {g[Y(t_1 ), \cdots ,Y(t_n )]} |\} < \infty $, we find \[ ({\text{i}})\qquad E\left\{ \left| {g\left[ Y\left( t_1 \right) , \cdots ,Y\left( t_n \right) \right]} \right| \theta (\omega ) = \theta _0 \right\} = E\left\{ g\left[ X\left( t_1 + \theta _0 \right) , \cdots ,X\left( t_n + \theta _0 \right) \right] \right\} \] for almost every $\theta _0 $ relative to $\mu $, the measure induced on the real line by $\theta (\omega )$. When $\theta $ 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...