On a modified version of the Lindley recursion

Abstract

This paper concerns a modified version of the Lindley recursion, where the recursion equation is given by $W_{i+1} = [V_{i} W_{i} + Y_{i}]^{+}$, with $\{V_i\}_{i=0}^{\infty }$ and $\{Y_i\}_{i=0}^{\infty }$ being two independent sequences of i.i.d. random variables. Additionally, we assume that the $V_i$ take values in $(-\infty , 1]$ and the $Y_i$ have a rational Laplace–Stieltjes transform. Under these assumptions, we investigate the transient and steady-state behaviors of the process $\{W_i\}_{i=0}^{\infty }$ by deriving an expression for the generating function of the Laplace–Stieltjes transform of the $W_i$.