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Consider a Markov chain $X_n$ on $\mathbb{Z}$ with transition probability matrix $S$. Let $C$ be a parameter representing the length of one step. We assume that in each step, we move to a point with distance of order $O(C)$, and the probability of move to each point is of order $C^{-1}$. This makes our Markov chain looks like a random walk. For a random walk, we have the local limit theorem on $P_x(X_n＝y)$. So Intuitively, we believe that there should at least be an exponentiallying decaying upper bound for $P_x(X_n＝y)$ with respect to $|x-y|/C$ in our setting. But I don't know how to prove this bound, or even the local limit theorem.

I want some references about this kind of estimate or local limit theorem.

Any help is appreciated, thanks in advance.

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  • $\begingroup$ What do you mean by "an exponential lying decaying upper bound"? $\endgroup$ Commented May 20 at 16:12
  • $\begingroup$ "exponential [...] with respect to $|x-y|/C$" -- What about the case $S_{i,i+1}=1$ for all $i\in\Bbb Z$? $\endgroup$ Commented May 20 at 16:14
  • $\begingroup$ @losif Pinelis: I apologize for the missspelling. I want to express exponentially, but it's automatically ''corrected''. $\endgroup$ Commented May 21 at 7:18
  • $\begingroup$ For the second question, I mean there should be a bound like $O(\exp(-|x-y|/C)$. But I think this bound should be also related to $n$. $\endgroup$ Commented May 21 at 7:23
  • $\begingroup$ What exactly the assumptions on the transition probabilities are? $\endgroup$ Commented May 21 at 9:45

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