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- W2088032987 abstract "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 Eleft{ left| {gleft[ Yleft( t_1 right) , cdots ,Yleft( t_n right) right]} right| theta (omega ) = theta _0 right} = Eleft{ gleft[ Xleft( t_1 + theta _0 right) , cdots ,Xleft( 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. If $theta $ is independent of a periodically nonstationary X and $mu $ is absolutely continuous with respect to Lebesgue measure, then Y is strictly stationary if and only if $mu $ 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 $theta $ is independent of $X(s)$ and $X(t)$ for every pair s, t, then $X(t + theta (omega ),omega )$ is given as a quadratic mean limit of $X(t + theta _n (omega ),omega )$, where the $theta _n (omega )$ are simple functions converging pointwise to $theta (omega )$. In this case, (i) holds at least for the first two moments of Y, that is, $begin{gathered} ({text{ii}})quad E{ Y(t)| {theta (omega ) = theta _0 } } = E{ X(t + theta _0 )} quad {text{a.e..}},(mu ), hfill E{ Y(s)Y(t)| {theta (omega ) = } theta _0 } = E{ X(s + theta _0 )X(t + theta _0 )} quad {text{a.e.}},(mu ). hfill end{gathered} $ If (ii) holds for $theta $ 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." @default.
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- W2088032987 date "1974-01-01" @default.
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- W2088032987 title "Stationarizing Properties of Random Shifts" @default.
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- W2088032987 doi "https://doi.org/10.1137/0126017" @default.
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