A multiplicative version of the Lindley recursion

Abstract

This paper presents an analysis of the stochastic recursion $$W_{i+1} = [V_iW_i+Y_i]^+$$ W i + 1 = [ V i W 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 $$Y_i=B_i-A_i$$ Y i = B i - A i , for independent sequences of nonnegative i.i.d. random variables $$\{A_i\}_{i\in {\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\{B_i\}_{i\in {\mathbb N}_0}$$ { B i } i ∈ N 0 , and assuming $$\{V_i\}_{i\in {\mathbb N}_0}$$ { V i } i ∈ N 0 is an i.i.d. sequence as well (independent of $$\{A_i\}_{i\in {\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\{B_i\}_{i\in {\mathbb N}_0}$$ { B i } i ∈ N 0 ), we then consider three special cases (i) $$V_i$$ V i equals a positive value a with certain probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) and is negative otherwise, and both $$A_i$$ A i and $$B_i$$ B i have a rational LST, (ii) $$V_i$$ V i attains negative values only and $$B_i$$ B i has a rational LST, (iii) $$V_i$$ V i is uniformly distributed on [0, 1], and $$A_i$$ 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.

Queueing and risk models with dependencies

“…x + denoting max(0, x) (similarly, below x − denotes min(0, x)). In the following, we assume that the steady-state waiting time distribution exists, which we claim to hold for any c > 0 as long as EB 1 < ∞ (and actually even if E log(1 + B 1 ) < ∞; see the account of this issue for a very similar model in [10]). Now we let i → ∞ in (3) and study the limiting random variable W ; let ω(s) denote the LST corresponding to W .…”

A dual risk model with additive and proportional gains: ruin probability and dividends

“…Palmowski et al [20] focus on a discrete-time set-up and study the finite-time ruin probability. In terms of analysis technique, the approach in Sections 2 and 3 bears similarities to the approach used in [12,10,11,23] to study Lindley-type recursions W n+1 = max(0, aW n + X n ), where a = 1 in the classical setting of a single-server queue with W n the waiting time of the nth customer.…”

A dual risk model with additive and proportional gains: ruin probability and dividends

“…Palmowski et al [20] focus on a discrete-time set-up and study the finite-time ruin probability. In terms of analysis technique, the approach in Sections 2 and 3 bears similarities to the approach used in [12,10,11,23] to study Lindley-type recursions W n+1 = max(0, aW n + X n ), where a = 1 in the classical setting of a single server queue with W n the waiting time of the nth customer.…”