Null Recurrent String

A.S. Gajrat, R. Iasnogorodski, V.A. Malyshev

1996, v.2, Issue 3, 427-460


A finite string $\alpha =a_1a_2\ldots a_n$ is a sequence of symbols from some alphabet $R=\{1,2,\ldots,r\}$. We define its Markovian evolution by some transition probabilities, dependent only on the right-most symbol, of erasing this symbol or of substituting it by two other symbols. In the case that such chains are null recurrent, we get limit laws for the distribution of the length of the string, of its right-most symbol and of the number of symbols $i$ in the string in the large time limit. Applications of these results are random walks on some non-commutative groups and queues with several customer types.

Keywords: random walk,stabilisation law,central limit theorem


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