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Боровков, А. А.; Mogulskii, A. A.

doi:10.1023/a:1004832928857

Cited by 11 publications

References 12 publications

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The Exact Asymptotics for Hitting Probability of a Remote Orthant by a Multivariate Lévy Process: The Cramér Case

Borovkov

Palmowski

2019

2017 MATRIX Annals

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For a multivariate Lévy process satisfying the Cramér moment condition and having a drift vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by the multivariate ruin problem introduced in F. Avram et al. (2008) in the two-dimensional case. Our solution relies on the analysis from Y. Pan and K. Borovkov (2017) for multivariate random walks and an appropriate time discretization.

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The Exact Asymptotics for Hitting Probability of a Remote Orthant by a Multivariate Lévy Process: The Cramér Case

Borovkov

Palmowski

2019

2017 MATRIX Annals

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Asymptotic Analysis of Random Walks: Light-Tailed Distributions

Боровков

Ulyanov

Zhitlukhin

2020

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This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time.

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The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem

Pan

Borovkov

2019

Adv. Appl. Probab.

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For a multivariate random walk with i.i.d. jumps satisfying the Cramér moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by A. Borovkov and A. Mogulskii from around 2000 with new auxiliary constructions, which enable us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a "corner" at the "most probable hitting point". We also discuss how our results can be extended to the case of more general target sets.

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Untitled

Cited by 11 publications

References 12 publications

The Exact Asymptotics for Hitting Probability of a Remote Orthant by a Multivariate Lévy Process: The Cramér Case

The Exact Asymptotics for Hitting Probability of a Remote Orthant by a Multivariate Lévy Process: The Cramér Case

Asymptotic Analysis of Random Walks: Light-Tailed Distributions

The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem

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