If Xt is strictly stationary, show that the joint distribution function of Xt1,...,Xtn depends only on...

If Xt is strictly stationary, show that the joint distribution function of Xt1,...,Xtn depends only on the time difference t2-t1,...,tn-t(n-1)

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The class of all stochastic processes is too large to work with in practice. We consider only the subclass of stationary processes.

{X_t} is said to be completely stationary if, for all n 1, for any t₁,t₂, . . . ,tn T,

and for any such that t₁+,t₂+ , . . . ,tn+ T are also contained in the index set,

the joint cdf of {X_t1 , X_t2 , . . . , X_tn } is the same as that of {X_t1+ , X_t2+ , . . . , X_tn+}

i.e., F_t1,_t2,_...,tn (a1, a2, . . . , an) = F_{t1+,t2+,...,tn+}(a1, a2, . . . , an),

so that the probabilistic structure of a completely stationary process is invariant under a shift in time

orchestra answered 1 year ago

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