## The Annals of Applied Probability

### Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities

Søren Asmussen

#### Abstract

Consider a reflected random walk $W_{n+1} = (W_n + X_n)^+$, where $X_0, X_1,\dots$ are i.i.d. with negative mean and subexponential with common distribution F. It is shown that the probability that the maximum within a regenerative cycle with mean $\mu$ exceeds x is approximately $\mu\bar{F}(x)$ as $x \to \infty$, and thereby that $\max(W_0, \dots, W_n)$ has the same asymptotics as $\max(X_0, \dots, X_n)$ as $n \to \infty$. In particular, the extremal index is shown to be $\theta = 0$, and the point process of exceedances of a large level is studied. The analysis extends to reflected Lévy processes in continuous time, say, stable processes. Similar results are obtained for a storage process with release rate $r(x)$ at level x and subexponential jumps (here the extremal index may be any value in $[0, \infty]$; also the tail of the stationary distribution is found. For a risk process with premium rate $r(x)$ at level x and subexponential claims, the asymptotic form of the infinite-horizon ruin probability is determined. It is also shown by example $[r(x) = a + bx$ and claims with a tail which is either regularly varying, Weibull- or lognormal-like] that this leads to approximations for finite-horizon ruin probabilities. Typically, the conditional distribution of the ruin time given eventual ruin is asymptotically exponential when properly normalized.

#### Article information

Source
Ann. Appl. Probab., Volume 8, Number 2 (1998), 354-374.

Dates
First available in Project Euclid: 9 August 2002

https://projecteuclid.org/euclid.aoap/1028903531

Digital Object Identifier
doi:10.1214/aoap/1028903531

Mathematical Reviews number (MathSciNet)
MR1624933

Zentralblatt MATH identifier
0942.60034

#### Citation

Asmussen, Søren. Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 (1998), no. 2, 354--374. doi:10.1214/aoap/1028903531. https://projecteuclid.org/euclid.aoap/1028903531

#### 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.
• 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.
• 3 ASMUSSEN, S. 1987. Applied Probability and Queues. Wiley, New York.
• 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.
• 5 ASMUSSEN, S. and KLUPPELBERG, C. 1995. Large deviations results in the presence of ¨ heavy 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 ¨ heavy tails. J. Appl. Probab. 34 208 212.
• 7 ASMUSSEN, S. and NIELSEN, H. M. 1995. Ruin probabilities via local adjustment coefficients. J. Appl. Probab. 32 736 755.
• 8 ASMUSSEN, S. and SCHOCK PETERSEN, S. 1989. Ruin probabilities expressed in terms of storage processes. Adv. in Appl. Probab. 20 913 916.
• 9 BALKEMA, A. A. and DE HAAN, L. 1974. Residual life-time at great age. Ann. Probab. 2 792 804.
• 10 BERMAN, S. M. 1962. Limit distribution of the maximum term in a sequence of dependent random variables. Ann. Math. Statist. 33 894 908.
• 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.
• 12 DASSIOS, A. and EMBRECHTS, P. 1989. Martingales and insurance risk. Stochastic Models 5 181 217.
• 13 DURRETT, R. 1980. Conditioned limit theorems for random walks with negative drift. Z. Wahrsch. Verw. Gebiete 52 277 287.
• 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.
• 15 EMBRECHTS, P., KLUPPELBERG, C. and MIKOSCH, T. 1997. Extremal Events in Finance and ¨ Insurance. 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.
• 17 GELUB, J. L. and DE HAAN, L. 1987. Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40. CWI, Amsterdam.
• 18 GNEDENKO, B. V. and KOVALENKO, I. N. 1989. Introduction to Queueing Theory, 2nd ed. Birkhauser, Basel. ¨
• 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.
• 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.
• 21 HARRISON, J. M. and RESNICK, S. I. 1977. The recurrence classification of risk and storage processes. Math. Oper. Res. 3 57 66.
• 22 IGLEHART, D. L. 1972. Extreme values in the GI G 1 queue. Ann. Math. Statist. 43 627 635.
• 23 KEILSON, J. 1979. Markov Chain Models Rarity and Exponentiality. Springer, New York.
• 24 KLUPPELBERG, C. 1988. Subexponential distributions and integrated tails. J. Appl. Probab. ¨ 25 132 141.
• 25 KLUPPELBERG, C. and STADTMULLER, U. 1995. Ruin probabilities in the presence of heavy¨ ¨ tails and interest rates. Scand. Actuar. J. 49 58.
• 26 LEADBETTER, M. R., LINDGREN, G. and ROOTZEN, H. 1983. Extremes and Related Properties ´ of Random Sequences and Processes. Springer, New York.
• 27 RESNICK, S. I. 1987. Extreme Values, Regular Variation Point and Processes. Springer, New York.
• 28 ROOTZEN, H. 1988. Maxima and exceedances of stationary Markov chains. Adv. in Appl. ´Probab. 20 371 390.
• 29 SUNDT, B. and TEUGELS, J. L. 1995. Ruin estimates under interest force. Insurance Math. Econom. 16 7 22.
• 30 SUNDT, B. and TEUGELS, J. L. 1996. The adjustment coefficient in ruin estimates under interest force. Insurance Math. Econom.