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A simple proof of heavy tail estimates for affine type Lipschitz recursions

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  • Buraczewski, Dariusz
  • Damek, Ewa

Abstract

We study the affine recursion Xn=AnXn−1+Bn where (An,Bn)∈R+×R is an i.i.d. sequence and recursions Xn=Φn(Xn−1) defined by Lipschitz transformations such that Φ(x)≥Ax+B. It is known that under appropriate hypotheses the stationary solution X has regularly varying tail, i.e. limt→∞tαP[X>t]=C. However positivity of C in general is either unknown or requires some additional involved arguments. In this paper we give a simple proof that C>0. This applies, in particular, to the case when Kesten–Goldie assumptions are satisfied.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:2:p:657-668
    DOI: 10.1016/j.spa.2016.06.022
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    References listed on IDEAS

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    1. Dyszewski, Piotr, 2016. "Iterated random functions and slowly varying tails," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 392-413.
    2. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
    3. Alsmeyer, Gerold, 2016. "On the stationary tail index of iterated random Lipschitz functions," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 209-233.
    4. 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.
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    Cited by:

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    2. 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.
    3. Damek, Ewa & Kołodziejek, Bartosz, 2020. "Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1792-1819.
    4. Yuri Kabanov & Sergey Pergamenshchikov, 2022. "On ruin probabilities with investments in a risky asset with a regime-switching price," Finance and Stochastics, Springer, vol. 26(4), pages 877-897, October.

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