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Tail estimates for stochastic fixed point equations via nonlinear renewal theory

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  • Collamore, Jeffrey F.
  • Vidyashankar, Anand N.

Abstract

This paper introduces a new approach, based on large deviation theory and nonlinear renewal theory, for analyzing solutions to stochastic fixed point equations of the form V=Df(V), where f(v)=Amax{v,D}+B for a random triplet (A,B,D)∈(0,∞)×R2. Our main result establishes the tail estimate P{V>u}∼Cu−ξ as u→∞, providing a new, explicit probabilistic characterization for the constant C. Our methods rely on a dual change of measure, which we use to analyze the path properties of the forward iterates of the stochastic fixed point equation. To analyze these forward iterates, we establish several new results in the realm of nonlinear renewal theory for these processes. As a consequence of our techniques, we develop a new characterization of the extremal index, as well as a Lundberg-type upper bound for P{V>u}. Finally, we provide an extension of our main result to random Lipschitz maps of the form Vn=fn(Vn−1), where fn=Df and Amax{v,D∗}+B∗≤f(v)≤Amax{v,D}+B.

Suggested Citation

  • Collamore, Jeffrey F. & Vidyashankar, Anand N., 2013. "Tail estimates for stochastic fixed point equations via nonlinear renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3378-3429.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:9:p:3378-3429
    DOI: 10.1016/j.spa.2013.04.015
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    1. Krishna Athreya, 2003. "Stationary measures for some Markov chain models in ecology and economics," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 23(1), pages 107-122, December.
    2. Borkovec, Milan, 2000. "Extremal behavior of the autoregressive process with ARCH(1) errors," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 189-207, February.
    3. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
    4. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
    5. Alsmeyer, Gerold & Fuh, Cheng-Der, 2001. "Limit theorems for iterated random functions by regenerative methods," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 123-142, November.
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    Cited by:

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    2. Damek, Ewa & Latała, Rafał & Nayar, Piotr & Tkocz, Tomasz, 2015. "Two-sided bounds for Lp-norms of combinations of products of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1688-1713.
    3. Buraczewski, Dariusz & Damek, Ewa, 2017. "A simple proof of heavy tail estimates for affine type Lipschitz recursions," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 657-668.
    4. janssen, Anja & Segers, Johan, 2013. "Markov Tail Chains," LIDAM Discussion Papers ISBA 2013017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.

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