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Suppose a random walk on an infinite line $[...-3,-2,-1,0,1,2,3,...]$, starting from 0. Probability to go right or left are equal. Does such a process stationary? I think that it is NOT, since the support of each step is different. I.e., $x_1\in{\{-1,1\}}, x_2\in{-2,0,2}, ...$. Thanks.

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    Got somethi$n$g from the answer below?2012-08-11

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After some clarifications in the comments, the question is whether such a process admits a stationary distribution. The answer is NO.

In fact, it appears that finding some good introductory textbook on random processes and pondering it quietly would save the OP some time. I suggest to try the (freely available on the web) first chapter of Markov chains by James Norris.