# Reference on a markov chain / Queue

Im looking for a reference that treats the Markov Chain defined by

$$W_i=(W_{i-1}-1)\vee X_i$$ where $$X_i\geq 0$$ are i.i.d discrete variables. In particular im interested in a reference that treats existence and moments of the invariant distribution, and gives moments of the return time to $$0$$ under some imposed assumptions on $$W_0$$ . I am not interested in a proof produced & presented here on overflow. Just a reference that i can put in my article.

This kind of process is sometimes called random exchange process. A starting point for a literature research might be the following article by Helland and Nilsen:

For a recent work with a characterization of nullrecurrence see

https://arxiv.org/abs/1608.01394

Addendum: Kellerer has studied order-preserving Markov chains in the following paper.

http://www.mathematik.uni-muenchen.de/~kellerer/0.pdf

The invariant distribution and its existence is independent of the starting distribution, at least under slight irreducibility conditions. In his comment after Theorem 10.1 Kellerer gives a characterization for finite first moment of an invariant distribution for exchange processes. For studying return times and their expectation values, his work in Chapter 11 might be helpful. For example, when $$X_1$$ is bounded, I believe that Theorem 11.2 and Proposition 11.3 directly imply finiteness of the expectation of all nondegenerate return times (possibly this is a rather trivial result). When $$X_1$$ is not bounded, things are more difficult of course.

• Thanks. After looking at material cited these, it seems that there are really not alot written about this chain. I find it strange. Would you know a reference for material treating the "in particular..." points in my question? Feb 28 '19 at 14:50

You can find the chain in Section 3 of Lamperti's work 'Maximal branching processes and ‘long-range percolation’' (see also the appendix by Kesten).

https://www.cambridge.org/core/journals/journal-of-applied-probability/article/abs/maximal-branching-processes-and-longrange-percolation/50B2EB4757739E6E2126DEE63B15D443

The recurrence properties (but not the moment of the return time) are collected in Proposition 3.6 here: https://arxiv.org/abs/2008.10585