A multiplicative version of the Lindley recursion

This paper presents an analysis of the stochastic recursion W_i+1=[V_iW_i+Y_i]⁺ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model’s stability condition. Writing Y_i=B_i−A_i, for independent sequences of nonnegative i.i.d. random variables {Ai}i_∈N0 and {B_i}_i∈N0, and assuming {V_i}_i∈N0 is an i.i.d. sequence as well (independent of {A_i}_i∈N0 and {B_i}_i∈N0), we then consider three special cases (i) V_i equals a positive value a with certain probability p∈(0,1) and is negative otherwise, and both A_i and B_i have a rational LST, (ii) V_i attains negative values only and B_i has a rational LST, (iii) V_i is uniformly distributed on [0, 1], and A_i is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.