The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem. (September 2019)

• Record Type: Journal Article
• Authors: Pan, Yuqing; Borovkov, Konstantin A.
• Is Part Of: Advances in applied probability. Volume 51:Number 3(2019)
• Page Start: 835
• Page End: 864
• Publication Date: 2019-09
• Subjects: Large deviation, -- exact asymptotics, -- multivariate random walk, -- multivariate ruin problem, -- Cramér moment condition, -- boundary crossing, -- second rate function; Primary 60F10, -- Secondary 60K35, -- 60G50; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2019.28
• Languages: English
• ISSNs: 0001-8678

Abstract

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.