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The supremum of a negative drift random walk with dependent heavy-tailed steps. (English) Zbl 1083.60506
Summary: Many important probabilistic models in queueing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk.When the stationary process is heavy-tailed, the corresponding ruin probabilities are high and the stationary distributions are heavy-tailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke.We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example, by a linear process.

60F10 Large deviations
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
Full Text: DOI Euclid
[1] Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for nonstandard risk and queueing processes withsubexponential jumps. Adv. Appl. Probab. 31 442-447. · Zbl 0942.60033 · doi:10.1239/aap/1029955142
[2] Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Palm-Martingale Calculus and Stochastic Recurrences. Springer, Berlin. · Zbl 0801.60081
[3] Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge Univ. Press. · Zbl 0617.26001
[4] Brockwell, P. and Davis, R. (1991). Time Series: Theory and Methods. Springer, New York, 2nd ed. · Zbl 0709.62080
[5] Chistyakov, V. (1964). A theorem on sums of independent random variables and its applications to branching random processes. Theory Probab. Appl. 9 640-648. Cline, D. (1983a). Estimation and linear prediction for regression, autoregression and ARMA withinfinite variance data. Ph.D. dissertation, Colorado State Univ. Cline, D. (1983b). Infinite series of random variables withregularly varying tails. Technical Report 83-24, Institute Applied Math. Statist., Univ. British Columbia, Vancouver, B.C. · Zbl 0203.19401
[6] Cline, D. and Hsing, T. (1991). Large deviation probabilities for sums and maxima of random variables withheavy or subexponential tails. Preprint, Texas A&M Univ.
[7] Cunha, C., Bestavros, A. and Crovella, M. (1995). Characteristics of WWW client-based traces. Preprint. Available as BU-CS-95-010 from crovella best cs.bu.edu.
[8] Davis, R. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed stationary processes withapplications to ARCH. Ann. Statist. 26 2049-2080. · Zbl 0929.62092 · doi:10.1214/aos/1024691368
[9] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179-195. · Zbl 0562.60026 · doi:10.1214/aop/1176993074
[10] Davis, R. and Resnick, S. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14 533-558. · Zbl 0605.62092 · doi:10.1214/aos/1176349937
[11] Davis, R. and Resnick, S. (1996). Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Probab. 6 1191-1210. · Zbl 0879.60053 · doi:10.1214/aoap/1035463328
[12] Embrechts, P., Kl üppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116
[13] 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 · doi:10.1016/0167-6687(82)90021-X
[14] Hsing, T., H üsler, J. and Leadbetter, M. (1988). On the exceedance point process for a stationary sequence. Probab. Theory Related Fields 78 97-112. · Zbl 0619.60054 · doi:10.1007/BF00718038
[15] Kokoszka, P. and Taqqu, M. (1996). Parameter estimation for infinite variance fractional ARIMA. Ann. Statist. 24 1880-1913. · Zbl 0896.62092 · doi:10.1214/aos/1069362302
[16] Kwapie ń, S. and Woyczy ński, N. (1992). RandomSeries and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston. Nagaev, A. (1969a). Integral limit theorems for large deviations when Cramér’s condition is not fulfilled I, II. Theory Probab. Appl. 14 51-64, 193-208. Nagaev, A. (1969b). Limit theorems for large deviations where Cramér’s conditions are violated. Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 6 17-22. (In Russian.)
[17] Nagaev, S. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745-789. · Zbl 0418.60033 · doi:10.1214/aop/1176994938
[18] Petrov, V. (1995). Limit Theorems of Probability Theory. Oxford Univ. Press. · Zbl 0826.60001
[19] Prokhorov, Y. (1959). An extremal problem in probability theory. Theory Probab. Appl. 4 201-204. · Zbl 0093.15102 · doi:10.1137/1104017
[20] Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York. · Zbl 0633.60001
[21] Resnick, S. (1997). Why non-linearities can ruin the heavy tailed modeler’s day. In A Practial Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions 219-240. Birkhäuser, Boston. · Zbl 0954.62107
[22] Resnick, S., Samorodnitsky, G. and Xue, F. (1999). How misleading can sample ACF’s of stable MA’s be? (Very!). Ann. Appl. Probab. 9 797-817. · Zbl 0959.62076 · doi:10.1214/aoap/1029962814
[23] Rosenblatt, M. (1962). RandomProcesses. Oxford Univ. Press. · Zbl 0111.32902 · doi:10.1137/0110008
[24] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 423-439. · Zbl 0135.18701 · doi:10.1214/aoms/1177700153
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