Stability of perpetuities in Markovian environment

Abstract

The stability of iterations of affine linear maps $\Psi_{n}(x)=A_{n}x+B_{n}$, $n=1,2,\ldots$, 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 $\pi$. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations $\Psi_{1}\circ\ldots\circ\Psi_{n}(Z_{0})$ 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 $\Psi_{n}\circ\ldots\circ\Psi_{1}$. The main differences caused by the Markovian environment as opposed to the extensively studied case of independent and identically distributed (iid) $\Psi_{1},\Psi_{2},\ldots$ 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, Grincevi\v{c}ius, and Goldie and Maller, in combination with recent results by the authors on fluctuation theory for Markov random walks.Comment: 36 page

Exponential functionals of Markov additive processes

“…Denoting the k-th return time of (J u n ) n∈N to j as τ u k (j), we note that τ u k (j) ℓ=1 A u ℓ = exp(−ξ τ u k (j)u ) does not tend to 0 a.s. for k → ∞ due to our assumption. Together with (4.22) we thus conclude from [1,Thm. 3.4] that |Z u n | P π u −→ ∞, n → ∞, where the invariant distribution π u of (J u n ) n∈N 0 is equivalent to π.…”

“…Remark 4.6. Note that uniqueness of the sequence {c j , j ∈ S} in (4.6) as shown in [1] directly implies uniqueness of the sequence {c j , j ∈ S} in (4.3).…”

“…sequences (A n , B n ), if almost sure convergence fails, the perpetuity (2.5) can still converge in probability. More precisely, from [1,Thm. 3.4] we derive under the assumption (2.6) that P j (Z n ∈ ·) converges weakly to some probability measure Q j if lim n→∞ τn(j) k=1 A k = 0 P π -a.s. and (2.9)…”

“…5.5 for more details). Then its stationary law π M is equivalent to π and as shown in [1,Eq. (17) and Lemma 4.1] degeneracy of the Markov modulated perpetuity (4.4) in the sense of (2.10) is equivalent to the existence of a unique sequence {c j , j ∈ S} in R such that…”

An arcsine law for Markov random walks

“…exists for some i ∈ S, then it does so and is the same for any i ∈ S (so we may replace P i with P π ), further satisfies ρ = lim n→∞ 1 n n k=1 P i (S τ k (i) > 0) (2) for all i ∈ S, and entails that an arcsine law holds for N > n := n k=1 1 {S k >0} and N n := n − N > n = n k=1 1 {S k ≤0} .…”