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Iterated random functions and slowly varying tails

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  • Dyszewski, Piotr

Abstract

Consider a sequence of i.i.d. random Lipschitz functions {Ψn}n≥0. Using this sequence we can define a Markov chain via the recursive formula Rn+1=Ψn+1(Rn). It is a well known fact that under some mild moment assumptions this Markov chain has a unique stationary distribution. We are interested in the tail behaviour of this distribution in the case when Ψ0(t)≈A0t+B0. We will show that under subexponential assumptions on the random variable log+(A0∨B0) the tail asymptotic in question can be described using the integrated tail function of log+(A0∨B0). In particular we will obtain new results for the random difference equation Rn+1=An+1Rn+Bn+1.

Suggested Citation

  • Dyszewski, Piotr, 2016. "Iterated random functions and slowly varying tails," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 392-413.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:2:p:392-413
    DOI: 10.1016/j.spa.2015.09.005
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    References listed on IDEAS

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    1. Maulik, Krishanu & Zwart, Bert, 2006. "Tail asymptotics for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 156-177, February.
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    Cited by:

    1. McKinlay, Shaun, 2017. "On beta distributed limits of iterated linear random functions," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 33-41.
    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. 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. Yang Yang & Shuang Liu & Kam Chuen Yuen, 2022. "Second-Order Tail Behavior for Stochastic Discounted Value of Aggregate Net Losses in a Discrete-Time Risk Model," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2600-2621, December.

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