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Pointwise estimates for first passage times of perpetuity sequences

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  • Buraczewski, D.
  • Damek, E.
  • Zienkiewicz, J.

Abstract

We consider first passage times τu=inf{n:Yn>u} for the perpetuity sequence Yn=B1+A1B2+⋯+(A1…An−1)Bn,where (An,Bn) are i.i.d. random variables with values in R+×R. Recently, a number of limit theorems related to τu were proved including the law of large numbers, the central limit theorem and large deviations theorems (see Buraczewski et al., in press). We obtain a precise asymptotics of the sequence P[τu=logu∕ρ], ρ>0, u→∞ which considerably improves the previous results of Buraczewski et al. (in press). There, probabilities P[τu∈Iu] were identified, for some large intervals Iu around ku, with lengths growing at least as loglogu. Remarkable analogies and differences to random walks (Buraczewski and Maślanka, in press; Lalley, 1984) are discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:9:p:2923-2951
    DOI: 10.1016/j.spa.2017.10.004
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    References listed on IDEAS

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    1. 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.
    2. 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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    6. Buraczewski, Dariusz, 2009. "On tails of fixed points of the smoothing transform in the boundary case," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3955-3961, November.
    7. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Cited by:

    1. Sebastian Mentemeier & Olivier Wintenberger, 2022. "Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(5), pages 750-780, September.

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