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Convergence to Stable Laws in the Space D

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

François Roueff  and
Philippe Soulier
François Roueff*
Affiliation:
Télécom ParisTech
Philippe Soulier*
Affiliation:
Laboratoire MODAL'X Université de Paris Ouest
*
∗ Postal address: Institut Mines-Télécom, Télécom ParisTech, CNRS LTCI-UMR5141, 46 rue Barrault, 75634 Paris Cedex 13, France.
∗∗ Postal address: Laboratoire MODAL'X Université de Paris Ouest, Nanterre, 92000, France. Email address: philippe.soulier@u-paris10.fr
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Abstract

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We study the convergence of centered and normalized sums of independent and identically distributed random elements of the space D of càdlàg functions endowed with Skorokhod's J1 topology, to stable distributions in D. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

Keywords

Regular variationstable processfunctional convergence

MSC classification

Primary: 60G52: Stable processes
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 1 - 17
Copyright
© Applied Probability Trust 

References

Basse-O'Connor, A. and Rosiński, J. (2013). On the uniform convergence of random series in Skorokhod space and representations of càdlàg infinitely divisible processes. Ann. Prob. 41, 4317–4341.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Davis, R. A. and Mikosch, T. (2008). Extreme value theory for space-time processes with heavy-tailed distributions. Stoch. Process. Appl. 118, 560–584.CrossRefGoogle Scholar
Davydov, Y. and Dombry, C. (2012). On the convergence of LePage series in Skorokhod space. Statist. Prob. Lett. 82, 145–150.CrossRefGoogle Scholar
Davydov, Y., Molchanov, I. and Zuyev, S. (2008). Strictly stable distributions on convex cones. Electron. J. Prob. 13, 259–321.CrossRefGoogle Scholar
de Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in C[0,1]. Ann. Prob. 29, 467–483.Google Scholar
Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274.CrossRefGoogle Scholar
Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N. S.) 80, 121–140.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principles Math. Sci. 288), 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699.CrossRefGoogle Scholar
Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.Google Scholar
Taqqu, M. S. and Levy, J. M. (1986). Using renewal processes to generate long-range dependence and high variability. In Dependence in Probability and Statistics. Birkhäuser, Boston, MA, pp. 73–89.CrossRefGoogle Scholar
Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.CrossRefGoogle Scholar