Short Communications

Exponentials and $R$-recurrent random walks on groups

M. G. Shur

Moscow State Institute of Electronics and Mathematics — Higher School of Economics

Abstract: On a locally compact group $E$ with a countable base we consider a right random walk $X$ which for some $r>0$ has a unique (up to a positive multiplier) $r$-invariant measure. If this measure obeys some weak restrictions, then the random walk $X$ corresponds to the single continuous exponential on $E$. From this we obtain that we can implement some $R$-recurrent (by Tweedie) random walk on the group $E$ only in the case when this group is recurrent and, moreover, when there exists a Harris recurrent random walk on it.

Keywords: $r$-invariant measure, $R$-recurrent walk on a group, random walk, Harris recurrent walk, exponential.

Received: 27.04.2015

DOI: 10.4213/tvp5075

 English version: Theory of Probability and its Applications, 2017, 61:3, 505–513

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