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Let $p(x,y)$ denote the transition probability of a markov chain. Similarly, let $p^n(x,y)$ be the n-step transition probability. My question is, is there a formal name for $S(x,y):=\sup_n p^n(x,y)$. Notice that if the Markov chain is irreducible, then $\lim_{n\rightarrow\infty}p^n(x,y)=\pi(y)$, where $\pi$ is the stationary distribution. Of course, the supremum usually occurs much earlier at some finite $n$. I would greatly appreciate any references to the usage of $S(x,y)$ in some recent papers.

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I don't see, what the fact that $p^n(x,y)\to \pi(y)$ with $n\to\infty$ tells us anything useful about the supremum. Especially, taking into account that the convergence is seldom monotonic. The notion of $S$ I've never met, so I wonder whether you've found it useful for some problems. – S.D. Dec 4 '12 at 7:27

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