Abstract
Let {Z}>o be a random walk with a negative drift and independent and identically distributed increments with heavytailed distribution, and let M = sup>0 Zn be its supremum. Asmussen and Kliippelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → oo, for the distribution of the quadruple that includes the time x  T(X) to exceed level x, position ZT at this time, position ZT i at the prior time, and the trajectory up to it (similar results were obtained for the CramerLundberg insurance risk process). We obtain here several extensions of this result to various regenerativetype models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of T. The class of models includes Markovmodulated models as particular cases. We also study fluid models, the BjorkGrandell risk process, give examples where the order of x is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).
Original language  English 

Pages (fromto)  136151 
Number of pages  16 
Journal  Journal of Applied Probability 
Volume  51 
Issue number  1 
DOIs  
Publication status  Published  1 Jan 2014 
Keywords
 BjorkGrandell model
 Breiman's theorem
 Conditioned limit theorem
 Markov modulation
 Mean excess function
 Random walk
 Regenerative process
 Regular variation
 Ruin time
 Subexponential distribution
ASJC Scopus subject areas
 Mathematics(all)
 Statistics and Probability
 Statistics, Probability and Uncertainty
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Sergey Foss
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Professor
Person: Academic (Research & Teaching)