Stability of perpetuities in Markovian environment

Abstract

The stability of iterations of affine linear maps ${\mathrm{\Psi }}_n(x)=A_{n}x+B_n$, $n=1,2,\dots$, is studied in the presence of a Markovian environment, more precisely, for the situation when ${(A_n,B_n)}_{n\ge 1}$ is modulated by an ergodic Markov chain ${(M_n)}_{n\ge 0}$ with countable state space $\mathcal{S}$ and stationary distribution $\mathit{\pi }$. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations ${\mathrm{\Psi }}_1\circ \dots \circ {\mathrm{\Psi }}_n(Z_0)$ 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 $\mathcal{S}$ to $\mathbb{R}$ rather than distributions on $\mathbb{R}$, 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 ${\mathrm{\Psi }}_n\circ \dots \circ {\mathrm{\Psi }}_1$. The main differences caused by the Markovian environment as opposed to the extensively studied case of independent and identically distributed (iid) ${\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2,\dots$ are that: (1) backward iterations may still converge in distribution if a.s. convergence fails, (2) the degenerate case when $A_1{c}_{M_1}+B_1=c_{M_0}$ a.s. for suitable constants $c_i$, $i\in \mathcal{S}$, is by far more complex than the degenerate case for iid $(A_n,B_n)$ when $A_{1}c+B_1=c$ a.s. for some $c\in \mathbb{R}$, and (3) forward and backward iterations generally have different laws given $M_0=i$ for $i\in \mathcal{S}$ 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.

Keywords:

• Random affine map
• Markov modulation
• iterated function system
• forward and backward iteration
• perpetuity
• a.s. and distributional convergence
• stochastic fixed-point equation

AMS Subject Classifications:

• 60J10
• 60H25
• 60J15
• 60K05
• 60K15

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Both authors were partially supported by the Deutsche Forschungsgemeinschaft [SFB 878].