Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant

TL;DR

This paper investigates conditions under which the initial state of a reflected process becomes irrelevant over time, showing that under certain conditions, the process converges to a stationary distribution regardless of initial conditions.

Contribution

It provides a new characterization and sufficient conditions for the asymptotic irrelevance of initial conditions in Skorohod reflection maps on the nonnegative orthant.

Findings

Initial condition effects vanish asymptotically under natural stability conditions.

Reflected Lévy and Markov additive processes have unique stationary distributions.

Convergence to stationarity occurs regardless of initial state under specified conditions.

Abstract

A reflection map, induced by the deterministic Skorohod problem on the nonnegative orthant, is applied to an $\mathbb{R}^n$ valued function $X$ on $[0,\infty)$ and then to $a+X$, where $a$ is a nonnegative constant vector. A question that has been open for over 15 years is under what conditions the difference between the two resulting regulated functions converges to zero for any choice of $a$ as time diverges. This in turn implies that if one imposes enough stochastic structure that ensures that the reflection map applied to a multidimensional process $X$ converges in distribution then it will also converge in distribution when it is applied to $\eta+X$ where $\eta$ is any almost surely finite valued random vector that may even depend on the process $X$. In this paper we obtain a useful equivalent characterization of this property. As a result we are able to identify a natural sufficient condition in terms of the given data $X$ and the constant routing matrix. A similar necessary condition is also indicated. A particular implication of our analysis is that under additional stochastic assumptions, asymptotic irrelevance of the initial condition does note require the existence of a stationary distribution. As immediate corollaries of our (and earlier) results we conclude that under the natural stability conditions, a reflected L\'evy process as well as Markov additive process has a unique stationary distribution and converges in distribution to this stationary distribution for every initial condition. Extensions of the sufficient condition are then developed for reflection maps with drift and routing coefficients that may be time and state dependent; some implications to multidimensional insurance models are briefly discussed.