Stability of perpetuities in Markovian environment

Alsmeyer G., Buckmann F.

Abstract

The stability of iterations of affine linear maps Ѱn(X), Anx + Bn,n = 1, 2, . . ., is studied in the presence of a Markovian environment, more precisely, for the situation when (An, Bn)n≥ is modulated by an ergodic Markov chain (Mn)n≥0 with countable state space S and stationary distribution π. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations Ѱ1 o...o Ѱn (Z0) and also describe all possible limit laws as solutions to a certain Markovian stochastic fixed-point equation. As a consequence of the random environment, these limit laws are stochastic kernels from S to R rather than distributions on R, thus reflecting their dependence on where the driving chain is started. We give also necessary and sufficient conditions for the distributional convergence of the forward iterations Ѱn o...o Ѱ1. The main differences caused by the Markovian environment as opposed to the extensively studied case of independent and identically distributed (iid) Ѱ1,Ѱ2... are that: (1) backward iterations may still converge in distribution if a.s. convergence fails, (2) the degenerate case when A1cM1 + B1 = cM0 a.s. for suitable constants ci , i ∈ S, is by far more complex than the degenerate case for iid when (An, Bn) a.s. for some c ∈ R, and (3) forward and backward iterations generally have different laws given M0 = i for i ∈ S so that the former ones need a separate analysis. Our proofs draw on related results for the iid-case, notably by Vervaat, Grincevičius, and Goldie and Maller, in combination with recent results by the authors on fluctuation theory for Markov random walks.