Random linear recursions with dependent coefficients

Ghosh, Arka P.; Hay, Diana; Hirpara, Vivek; Rastegar, Reza; Roitershtein, Alexander; Schulteis, Ashley; Suh, Jiyeon

doi:10.1016/j.spl.2010.06.013

Cited by 10 publications

(25 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof combines ideas developed in [20,41,18]. We notice that Grey conjectured in [20] that using his method it may be possible to extend the results of [41] and rid of the assumption that (Q n ) n∈Z and (M n ) n∈Z are independent.…”

Section: Resultsmentioning

confidence: 90%

“…The next lemma, which generalizes Proposition 2.1 of [18], is the key element of our proof of Theorem 1.…”

Section: Appendix Proof Of Theoremmentioning

confidence: 88%

“…We remark that in dimension one the results of [27,19] (where M n is dominant in determining the tail behavior of X) and [20,23] (where Q n is dominant) were extended to a Markovian setup in [11,42,18], respectively.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multivariate linear recursions with Markov-dependent coefficients

Hay

Rastegar

Roitershtein

2011

Journal of Multivariate Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Resultsmentioning

confidence: 90%

“…The next lemma, which generalizes Proposition 2.1 of [18], is the key element of our proof of Theorem 1.…”

Section: Appendix Proof Of Theoremmentioning

confidence: 88%

See 1 more Smart Citation

Multivariate linear recursions with Markov-dependent coefficients

Hay

Rastegar

Roitershtein

2011

Journal of Multivariate Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…The degeneracy condition (16) In the iid-case, degeneracy occurs when P(Ac + B = c) = 1 for some c ∈ R and takes a side note only to deal with. This is quite different in the presence of a Markovian environment, and this subsection therefore collects some relevant facts about the degeneracy condition (16) and particularly shows its equivalence with (17). To avoid trivialities, we assume |S| > 1 throughout.…”

Section: If Condition (18) Failsmentioning

confidence: 99%

“…and this forces c i to be unique. (b) Turning to (16), pick any i, j ∈ S such that p ij > 0. Then P i (E n ) > 0 for all n ≥ 1, where…”

Section: If Condition (18) Failsmentioning

confidence: 99%

Stability of perpetuities in Markovian environment

Alsmeyer

Buckmann

2017

Journal of Difference Equations and Applications

View full text Add to dashboard Cite

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

show abstract

An impossibility theorem for wealth in heterogeneous-agent models with limited heterogeneity

Stachurski

Toda

2019

Journal of Economic Theory

View full text Add to dashboard Cite

It has been conjectured that canonical Bewley-Huggett-Aiyagari heterogeneous-agent models cannot explain the joint distribution of income and wealth. The results stated below verify this conjecture and clarify its implications under very general conditions. We show in particular that if (i) agents are infinitely-lived, (ii) saving is risk-free, and (iii) agents have constant discount factors, then the wealth distribution inherits the tail behavior of income shocks (e.g., light-tailedness or the Pareto exponent). Our restrictions on utility require only that relative risk aversion is bounded, and a large variety of income processes are admitted. Our results show conclusively that it is necessary to go beyond standard models to explain the empirical fact that wealth is heavier-tailed than income. We demonstrate through examples that relaxing any of the above three conditions can generate Pareto tails.

show abstract

Random linear recursions with dependent coefficients

Abstract: a b s t r a c tWe consider the equation R n = Q n + M n R n−1 , with random non-i.i.d. coefficients (Q n , M n ) n∈Z ∈ R 2 , and show that the distribution tails of the stationary solution to this equation are regularly varying at infinity.

Cited by 10 publications

References 33 publications

Multivariate linear recursions with Markov-dependent coefficients

Multivariate linear recursions with Markov-dependent coefficients

Stability of perpetuities in Markovian environment

An impossibility theorem for wealth in heterogeneous-agent models with limited heterogeneity

Contact Info

Product

Resources

About