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

Part of: Stochastic processes Special processes Limit theorems

Published online by Cambridge University Press:  03 September 2019

Yuqing Pan  and
Konstantin A. Borovkov
Yuqing Pan*
Affiliation:
The University of Melbourne
Konstantin A. Borovkov*
Affiliation:
The University of Melbourne
*
*Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville VIC 3010, Australia.
*Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville VIC 3010, Australia.
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Abstract

For a multivariate random walk with independent and identically distributed 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 of Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by Borovkov and Mogulskii from around 2000 with new auxiliary constructions, enabling 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.

Keywords

Large deviationexact asymptoticsmultivariate random walkmultivariate ruin problemCramér moment conditionboundary crossingsecond rate function

MSC classification

Primary: 60F10: Large deviations
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60G50: Sums of independent random variables; random walks
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 3 , September 2019 , pp. 835 - 864
Copyright
© Applied Probability Trust 2019 

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