On a class of multiplicative Lindley-type recursions with Markov-modulated dependencies

Abstract

In this paper, we study Markov-modulated dependencies for the multiplicative Lindley's recursion $W_{n+1}=[V_{n}W_{n}+Y_{n}(V_{n})]^{+}$, where $Y_{n}(V_{n})$ may depend on $V_{n}$, and can be written as the difference of two nonnegative random variables that also depend on a common background discrete-time Markov chain $\{Z_{n}\}_{n\in\mathbb{N}}$. Given the state of the background Markov chain, we consider two cases: a) $V_{n}$ equals either 1, or $a\in(0,1)$, or it is negative with certain probabilities, and $Y_{n}(V_{n}):=Y_{n}=S_{n}-A_{n+1}$, where both $A_n$ and $S_n$ have a rational Laplace-Stieltjes transform (LST). b) $V_{n}$ equals $1$ or $-1$ according to certain probabilities, and $Y_{n}(V_{n})$ follow a more general scheme, dependent on $V_{n}$. In both cases, we derive the LST of the stationary transform vector of $\{W_{n}\}_{n\in\mathbb{N}_{0}}$. In the second case, we also provide a recursive approach to obtain the steady-state moments and investigate its asymptotic behavior. A simple numerical example illustrates the theoretical findings.