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First passage times of general sequences of random vectors: A large deviations approach

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  • Collamore, Jeffrey F.

Abstract

Suppose is a sequence of random variables such that the probability law of Yn/n satisfies the large deviation principle and suppose . Let T(A)=inf{n: Yn[set membership, variant]A} be the first passage time and, to obtain a suitable scaling, let T[var epsilon](A)=[var epsilon]inf{n: Yn[set membership, variant]A/[var epsilon]}. We consider the asymptotic behavior of T[var epsilon](A) as [var epsilon]-->0. We show that the the probability law of T[var epsilon](A) satisfies the large deviation principle; in particular, as [var epsilon]-->0, where IA(·) is a large deviation rate function and C is any open or closed subset of [0,[infinity]). We then establish conditional laws of large numbers for the normalized first passage time T[var epsilon](A) and normalized first passage place Y[var epsilon]T[var epsilon](A).

Suggested Citation

  • Collamore, Jeffrey F., 1998. "First passage times of general sequences of random vectors: A large deviations approach," Stochastic Processes and their Applications, Elsevier, vol. 78(1), pages 97-130, October.
    • Handle: RePEc:eee:spapps:v:78:y:1998:i:1:p:97-130
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    Cited by:

    1. Anita Behme & Philipp Lukas Strietzel, 2021. "A $$2~{\times }~2$$ 2 × 2 random switching model and its dual risk model," Queueing Systems: Theory and Applications, Springer, vol. 99(1), pages 27-64, October.
    2. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
    3. Barbe, Ph. & McCormick, W.P., 2010. "An extension of a logarithmic form of Cramér's ruin theorem to some FARIMA and related processes," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 801-828, June.
    4. Albrecher, Hansjörg & Cheung, Eric C.K. & Liu, Haibo & Woo, Jae-Kyung, 2022. "A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 96-118.
    5. Gong, Lan & Badescu, Andrei L. & Cheung, Eric C.K., 2012. "Recursive methods for a multi-dimensional risk process with common shocks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 109-120.

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