Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities. (English) Zbl 0942.60034

Asmussen, Søren

Ann. Appl. Probab. 8, No. 2, 354-374 (1998).

Under the assumption that the jumps are iid and follow a subexponential distribution \( F \), three stochastic models are discussed: For the Lindley process \( (W_n)\) given by the recursion \( W_{n+1}=(W_n + X_n)^+ \) let \( M_{\tau} \) be the maximum within a regenerative cycle with mean \(\mu\) and let \(G(x)= P(M_{\tau}\leq x) \). It is shown that the assumption of subexponentiality results in the asymptotics \({\overline G}(x) \sim \mu{\overline F}(x)\) (which implies that the extremal index \(\theta\) of \( (W_n) \) is zero) and \(\|P(\max_{0\leq k\leq n} W_k \leq x) - G^{n/\mu}(x) \|\rightarrow 0\), \(n\rightarrow \infty \). If in addition \(F\) belongs to the max-domain of attraction of an extreme value df \(H\), then the point process of the exceedances, properly normalized, converges in distribution to a compound Poisson process with intensity \(-\log H\) and a Pareto compounding distribution. Similar results are obtained for a storage process \((V_t)\) which moves between the heavy-tailed jumps downwards according to the ODE \( {\dot x}(t)=-r(x(t))\) where \(r(x)\) is the release rate at level \(x\): the maximum of \((V_t)\) up to time \(T\) behaves like the maximal jump in \([0,T]\). The tail of the stationary distribution is also found. For a risk process with premium rate \(r(x)\) and subexponential claims the asymptotic distribution of the ruin time \(\rho (x)\) is determined: the conditional distribution of \(\rho (x)\) given \(\rho (x) < \infty\) converges to the exponential distribution.

Reviewer: E.Pancheva (Sofia)

MSC:

60G70	Extreme value theory; extremal stochastic processes
60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60K25	Queueing theory (aspects of probability theory)

Keywords:

cycle maximum; extreme values; ruin probability; storage process; heavy tails

References:

[1]	ANANTHARAM, V. 1988. How large delay s build up in a GI GI 1 queue. Queueing Sy stems Theory Appl. 5 345 368. · Zbl 0695.60092
[2]	ASMUSSEN, S. 1982. Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI G 1 queue. Adv. in Appl. Probab. 14 143 170. JSTOR: · Zbl 0501.60076
[3]	ASMUSSEN, S. 1987. Applied Probability and Queues. Wiley, New York. · Zbl 0624.60098
[4]	ASMUSSEN, S. 1996. Rare events in the presence of heavy tails. In Stochastic Networks: Z. Rare Events and Stability P. Glasserman, K. Sigman and D. Yao, eds. 197 214. Springer, New York. · Zbl 0856.60101
[5]	ASMUSSEN, S. and KLUPPELBERG, C. 1995. Large deviations results in the presence of ḧeavy tails, with applications to insurance risk. Stochastic Process. Appl. 64 103 125.
[6]	ASMUSSEN, S. and KLUPPELBERG, C. 1997. Stationary M G 1 excursions in the presence of ḧeavy tails. J. Appl. Probab. 34 208 212. JSTOR: · Zbl 0876.60080
[7]	ASMUSSEN, S. and NIELSEN, H. M. 1995. Ruin probabilities via local adjustment coefficients. J. Appl. Probab. 32 736 755. JSTOR: · Zbl 0834.60099
[8]	ASMUSSEN, S. and SCHOCK PETERSEN, S. 1989. Ruin probabilities expressed in terms of storage processes. Adv. in Appl. Probab. 20 913 916. JSTOR: · Zbl 0657.60111
[9]	BALKEMA, A. A. and DE HAAN, L. 1974. Residual life-time at great age. Ann. Probab. 2 792 804. · Zbl 0295.60014
[10]	BERMAN, S. M. 1962. Limit distribution of the maximum term in a sequence of dependent random variables. Ann. Math. Statist. 33 894 908. · Zbl 0109.11804
[11]	BROCKWELL, P. J., RESNICK, S. I. and TWEEDIE, R. L. 1982. Storage processes with general release rule and additive inputs. Adv. in Appl. Probab. 14 392 433. JSTOR: · Zbl 0482.60087
[12]	DASSIOS, A. and EMBRECHTS, P. 1989. Martingales and insurance risk. Stochastic Models 5 181 217. · Zbl 0676.62083
[13]	DURRETT, R. 1980. Conditioned limit theorems for random walks with negative drift. Z. Wahrsch. Verw. Gebiete 52 277 287. · Zbl 0416.60021
[14]	EMBRECHTS, P. and GOLDIE, C. M. 1980. On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29 243 256. · Zbl 0425.60011
[15]	EMBRECHTS, P., KLUPPELBERG, C. and MIKOSCH, T. 1997. Extremal Events in Finance and Ïnsurance. Springer, New York.
[16]	EMBRECHTS, P. and VERAVERBEKE, N. 1982. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55 72. · Zbl 0518.62083
[17]	GELUB, J. L. and DE HAAN, L. 1987. Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40. CWI, Amsterdam. · Zbl 0624.26003
[18]	GNEDENKO, B. V. and KOVALENKO, I. N. 1989. Introduction to Queueing Theory, 2nd ed. Birkhauser, Basel. \" · Zbl 0624.60108
[19]	GOLDIE, C. and RESNICK, S. I. 1988. Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution. Adv. in Appl. Probab. 20 706 718. JSTOR: · Zbl 0659.60028
[20]	HARRISON, J. M. and RESNICK, S. I. 1976. The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Oper. Res. 1 347 358. JSTOR: · Zbl 0381.60092
[21]	HARRISON, J. M. and RESNICK, S. I. 1977. The recurrence classification of risk and storage processes. Math. Oper. Res. 3 57 66. JSTOR: · Zbl 0397.90033
[22]	IGLEHART, D. L. 1972. Extreme values in the GI G 1 queue. Ann. Math. Statist. 43 627 635. · Zbl 0238.60072
[23]	KEILSON, J. 1979. Markov Chain Models Rarity and Exponentiality. Springer, New York. · Zbl 0411.60068
[24]	KLUPPELBERG, C. 1988. Subexponential distributions and integrated tails. J. Appl. Probab. \" 25 132 141. JSTOR: · Zbl 0651.60020
[25]	KLUPPELBERG, C. and STADTMULLER, U. 1995. Ruin probabilities in the presence of heavy\" ẗails and interest rates. Scand. Actuar. J. 49 58. · Zbl 1022.60083
[26]	LEADBETTER, M. R., LINDGREN, G. and ROOTZEN, H. 1983. Extremes and Related Properties óf Random Sequences and Processes. Springer, New York. · Zbl 0518.60021
[27]	RESNICK, S. I. 1987. Extreme Values, Regular Variation Point and Processes. Springer, New York. · Zbl 0633.60001
[28]	ROOTZEN, H. 1988. Maxima and exceedances of stationary Markov chains. Adv. in Appl. Ṕrobab. 20 371 390. JSTOR: · Zbl 0654.60023
[29]	SUNDT, B. and TEUGELS, J. L. 1995. Ruin estimates under interest force. Insurance Math. Econom. 16 7 22. · Zbl 0838.62098
[30]	SUNDT, B. and TEUGELS, J. L. 1996. The adjustment coefficient in ruin estimates under interest force. Insurance Math. Econom.