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

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)
Full Text:

### 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.