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Asymptotics for exponential functionals of random walks

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  • Xu, Wei

Abstract

This paper provides a detailed description for the asymptotics of exponential functionals of random walks with light/heavy tails. We give the convergence rate based on the key observation that the asymptotics depends on the sample paths with either slowly decreasing local minimum or final value below a low level. Also, our thoughtful analysis of the interrelationship between the local minimum and the final value provides the exact expression for the limiting coefficients in terms of some transformations of the random walk.

Suggested Citation

  • Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.
    • Handle: RePEc:eee:spapps:v:165:y:2023:i:c:p:1-42
    DOI: 10.1016/j.spa.2023.07.013
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    References listed on IDEAS

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    1. Barker, A. & Savov, M., 2021. "Bivariate Bernstein–gamma functions and moments of exponential functionals of subordinators," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 454-497.
    2. V. I. Afanasyev & C. Böinghoff & G. Kersting & V. A. Vatutin, 2012. "Limit Theorems for Weakly Subcritical Branching Processes in Random Environment," Journal of Theoretical Probability, Springer, vol. 25(3), pages 703-732, September.
    3. Søren Asmussen & Serguei Foss & Dmitry Korshunov, 2003. "Asymptotics for Sums of Random Variables with Local Subexponential Behaviour," Journal of Theoretical Probability, Springer, vol. 16(2), pages 489-518, April.
    4. Hui He & Zenghu Li & Wei Xu, 2018. "Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1952-1974, December.
    5. Afanasyev, V.I. & Geiger, J. & Kersting, G. & Vatutin, V.A., 2005. "Functional limit theorems for strongly subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1658-1676, October.
    6. Li, Zenghu & Xu, Wei, 2018. "Asymptotic results for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 108-131.
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