2017
DOI: 10.1080/10236198.2016.1271878
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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… Show more

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Cited by 6 publications
(23 citation statements)
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“…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.…”
Section: Proof Of Theorem 41: Divergencesupporting
confidence: 73%
See 3 more Smart Citations
“…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.…”
Section: Proof Of Theorem 41: Divergencesupporting
confidence: 73%
“…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).…”
Section: Degeneracy Of E (ξη) (T)mentioning
confidence: 93%
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“…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} .…”
Section: Introductionmentioning
confidence: 99%