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Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions

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  • Damek, Ewa
  • Kołodziejek, Bartosz

Abstract

We study the stochastic recursion Xn=Ψn(Xn−1), where (Ψn)n≥1 is a sequence of i.i.d. random Lipschitz mappings close to the random affine transformation x↦Ax+B. We describe the tail behaviour of the stationary solution X under the assumption that there exists α>0 such that E|A|α=1 and the tail of B is regularly varying with index −α<0. We also find the second order asymptotics of the tail of X when Ψ(x)=Ax+B.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:3:p:1792-1819
    DOI: 10.1016/j.spa.2019.05.016
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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. 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.
    3. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
    4. 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.
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    Cited by:

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