# stationary process is invariant under sliding time

$(\Omega,\Im,P)$ is a probability space and $\xi(t)\equiv\xi(w,t)$ is a stochastic process which is defined on $\Omega\times T$ ,$T=[0,\infty)$. If for every $0=t_{0}<t_{1}<...<t_{n}<\infty$ , $n=0,1,2,\ldots,x_{1},\ldots x_{n}\in\mathbb{R}$ and for all $h\in\mathbb{R}$ $$F_{t_{1},\ldots,t_{n}}(x_{1},...,x_{n})\equiv P(\xi(t_{1})\leq x_{1},\ldots,\xi(t_{n})\leq x_{n})\\\ \equiv P(\xi(t_{1}+h)\leq x_{1},\ldots,\xi(t_{n}+h)\leq x_{n})$$ then the process $\xi(t)$ is stationary process in narrow sense. In two dimensions distribution function doesn't change when the time is slided. How can I prove that?

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What do you want to prove exactly? –  Davide Giraudo Mar 24 '12 at 12:49
To expand on @Davide's comment: you state a definition then you ask to prove it. But one cannot prove a definition. –  Did Mar 24 '12 at 12:58
Of course i am not want to prove definition. How can i show that under sliding time distribution function doesn't change? I hope that i mean exactly what i want. –  demir Mar 24 '12 at 15:20
Sliding time sounds like changing $t_1,\dots,t_n$ to $t_1+h,\dots,t_n+h$, and remaining the same for distribution function means $F_{t_1,\dots,t_n}(x_1,\dots,x_n)=F_{t_1+h,\dots,t_n+h}(x_1,\dots,x_n)$ - definition. So maybe try to formulate your problem in math symbols –  Julius Mar 24 '12 at 15:33
You are right. Thank you. –  demir Mar 24 '12 at 16:12