## The Annals of Applied Probability

### On overload in a storage model, with a self-similar and infinitely divisible input

#### Abstract

Let {X(t)}t0 be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index H>0. Pick constants γ>H and c>0. Let ν be the Lévy measure on ℝ[0,) of X, and suppose that R(u)ν({y[0,):sup t0y(t)/(1+ctγ)>u}) is suitably “heavy tailed” as u (e.g., subexponential with positive decrease). For the “storage process” Y(t)sup st(X(s)X(t)c(st)γ), we show that P{sup s[0,t(u)]Y(s)>u}P{Y({}(u))>u} as u, when 0(u)t(u) do not grow too fast with u [e.g., t(u)=o(u1/γ)].

#### Article information

Source
Ann. Appl. Probab., Volume 14, Number 2 (2004), 820-844.

Dates
First available in Project Euclid: 23 April 2004

https://projecteuclid.org/euclid.aoap/1082737113

Digital Object Identifier
doi:10.1214/105051604000000125

Mathematical Reviews number (MathSciNet)
MR2052904

Zentralblatt MATH identifier
1047.60034

#### Citation

Albin, J. M. P.; Samorodnitsky, Gennady. On overload in a storage model, with a self-similar and infinitely divisible input. Ann. Appl. Probab. 14 (2004), no. 2, 820--844. doi:10.1214/105051604000000125. https://projecteuclid.org/euclid.aoap/1082737113

#### References

• Albin, J. M. P. (1998). Extremal theory for self-similar processes. Ann. Probab. 26 743--793.
• Albin, J. M. P. (1999). Extremes of totally skewed $\alpha$-stable processes. Stochastic Process. Appl. 79 185--212.
• Bingham, N. H. (1986). Variants on the law of the iterated logarithm. Bull. London Math. Soc. 18 433--467.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
• Burnecki, K., Rosiński, J. and Weron, A. (1998). Spectral representation and structure of stable self-similar processes. In Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis 1943--1995 (I. Karatzas, B. S. Rajput and M. S. Taqqu, eds.). Birkhäuser, Boston.
• Dacorogna, M. M., Gençay, R., Müller, U. A. and Pictet, O. V. (2001). Effective return, risk aversion and drawdowns. Phys. A 289 229--248.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 2, 2nd ed. Wiley, New York.
• Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83 257--271.
• Kôno, N. and Maejima, M. (1991). Self-similar stable processes with stationary increments. In Stable Processes and Related Topics (S. Cambanis, G. Samorodnitsky and M. S. Taqqu, eds.). Birkhäuser, Boston.
• Lamperti, J. W. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104 62--78.
• Maruyama, G. (1970). Infinitely divisible processes. Theory Probab. Appl. 15 1--22.
• Norros, I. (1994). A storage model with self-similar input. Queuing Systems 16 387--396.
• Pipiras, V. and Taqqu, M. S. (2002a). Decomposition of self-similar stable mixing moving averages. Probab. Theory Related Fields 123 412--452.
• Pipiras, V. and Taqqu, M. S. (2002b). The structure of self-similar stable mixing moving averages. Ann. Probab. 30 898--932.
• Piterbarg, V. I. (2001). Large deviations of a storage process with fractional Brownian motion as input. Extremes 4 147--164.
• Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinite divisible processes. Probab. Theory Related Fields 82 451--487.
• Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996--1014.
• Samorodnitsky, G. (1988). Extrema of skewed stable processes. Stochastic Process. Appl. 30 17--39.
• Samorodnitsky, G. and Taqqu, M. S. (1990). $(1/\alpha)$-self-similar processes with stationary increments. J. Multivariate Anal. 35 308--313.
• Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, London.
• Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
• Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1998). On the mixing structure of stationary increments and self-similar $\alpha$ processes. Unpublished manuscript.
• Talagrand, M. (1988). Small tails for the supremum of a Gaussian process. Ann. Inst. H. Poincaré Sect. B 24 307--315.