# zbMATH — the first resource for mathematics

On overload in a storage model, with a self-similar and infinitely divisible input. (English) Zbl 1047.60034
Summary: Let $$\{X(t)\}_{t\geq 0}$$ be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index $$H>0$$. Pick constants $$\gamma>H$$ and $$c>0$$. Let $$\nu$$ be the Lévy measure on $$\mathbb R^{[0,\infty)}$$ of $$X$$, and suppose that $R(u)\equiv \nu\biggl( \bigl\{y \in\mathbb R^{[0,\infty)}: \sup_{t\geq 0}y(t)/(1+ ct^\gamma) >u\bigr\}\biggr)$ is suitably “heavy tailed” as $$u\to\infty$$ (e.g., subexponential with positive decrease). For the “storage process” $$Y(t)\equiv\sup_{s\geq t}(X(s)-X(t)-c(s-t)^\gamma)$$, we show that ${\mathbf P}\Bigl\{\sup_{s\in [0,t(u)]} Y(s)>u\Bigr\} \sim{\mathbf P}\bigl\{Y(\widehat t(u))>u\bigr\} \text{ as }u\to\infty,$ when $$0 \leq\widehat t(u)\leq t(u)$$ do not grow too fast with $$u$$ [e.g., $$t(u)=o (u^{1/ \gamma})]$$.

##### MSC:
 60G18 Self-similar stochastic processes 60E07 Infinitely divisible distributions; stable distributions 60G10 Stationary stochastic processes 60G70 Extreme value theory; extremal stochastic processes 90B05 Inventory, storage, reservoirs
Full Text:
##### References:
  Albin, J. M. P. (1998). Extremal theory for self-similar processes. Ann. Probab. 26 743–793. · Zbl 0937.60033  Albin, J. M. P. (1999). Extremes of totally skewed $$\alpha$$-stable processes. Stochastic Process. Appl. 79 185–212. · Zbl 0961.60025  Bingham, N. H. (1986). Variants on the law of the iterated logarithm. Bull. London Math. Soc. 18 433–467. · Zbl 0633.60042  Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press. · Zbl 0617.26001  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. · Zbl 0915.60052  Dacorogna, M. M., Gençay, R., Müller, U. A. and Pictet, O. V. (2001). Effective return, risk aversion and drawdowns. Phys. A 289 229–248. · Zbl 0971.91510  Feller, W. (1971). An Introduction to Probability Theory and Its Applications , 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003  Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83 257–271. · Zbl 0997.60057  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. · Zbl 0728.60041  Lamperti, J. W. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104 62–78. · Zbl 0286.60017  Maruyama, G. (1970). Infinitely divisible processes. Theory Probab. Appl. 15 1–22. · Zbl 0268.60036  Norros, I. (1994). A storage model with self-similar input. Queuing Systems 16 387–396. · Zbl 0811.68059  Pipiras, V. and Taqqu, M. S. (2002a). Decomposition of self-similar stable mixing moving averages. Probab. Theory Related Fields 123 412–452. · Zbl 1007.60026  Pipiras, V. and Taqqu, M. S. (2002b). The structure of self-similar stable mixing moving averages. Ann. Probab. 30 898–932. · Zbl 1016.60057  Piterbarg, V. I. (2001). Large deviations of a storage process with fractional Brownian motion as input. Extremes 4 147–164. · Zbl 1003.60053  Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinite divisible processes. Probab. Theory Related Fields 82 451–487. · Zbl 0659.60078  Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996–1014. JSTOR: · Zbl 0776.60049  Samorodnitsky, G. (1988). Extrema of skewed stable processes. Stochastic Process. Appl. 30 17–39. · Zbl 0656.60029  Samorodnitsky, G. and Taqqu, M. S. (1990). $$(1/\alpha)$$-self-similar processes with stationary increments. J. Multivariate Anal. 35 308–313. · Zbl 0721.60047  Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes . Chapman and Hall, London. · Zbl 0925.60027  Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press. · Zbl 0973.60001  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. · Zbl 0641.60044
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.