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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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    Cited by:

    1. Buraczewski, D. & Damek, E. & Zienkiewicz, J., 2018. "Pointwise estimates for first passage times of perpetuity sequences," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2923-2951.
    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. 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.
    4. 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.
    5. 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).

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