Roueff, François; Soulier, Philippe Convergence to stable laws in the space \(\mathcal D\). (English) Zbl 1326.60039 J. Appl. Probab. 52, No. 1, 1-17 (2015). Summary: We study the convergence of centered and normalized sums of independent and identically distributed random elements of the space \(\mathcal D\) of càdlàg functions, endowed with Skorokhod’s \(J_{1}\) topology, to stable distributions in \(\mathcal D\). 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. Cited in 2 Documents MSC: 60F17 Functional limit theorems; invariance principles 60G52 Stable stochastic processes 60E07 Infinitely divisible distributions; stable distributions 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60K05 Renewal theory 60G70 Extreme value theory; extremal stochastic processes Keywords:functional convergence; stable process; Skorokhod \(J_{1}\) topology; regular variation; point process convergence; empirical process; renewal-reward process × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] 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. · Zbl 1287.60055 · doi:10.1214/12-AOP783 [2] Billingsley, P. (1968). Convergence of Probability Measures . John Wiley, New York. · Zbl 0172.21201 [3] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes , Vol. II, 2nd edn. Springer, New York. · Zbl 1159.60003 · doi:10.1007/978-0-387-49835-5 [4] Davis, R. A. and Mikosch, T. (2008). Extreme value theory for space-time processes with heavy-tailed distributions. Stoch. Process. Appl. 118, 560-584. · Zbl 1142.60040 · doi:10.1016/j.spa.2007.06.001 [5] Davydov, Y. and Dombry, C. (2012). On the convergence of LePage series in Skorokhod space. Statist. Prob. Lett. 82, 145-150. · Zbl 1230.60013 · doi:10.1016/j.spl.2011.09.011 [6] Davydov, Y., Molchanov, I. and Zuyev, S. (2008). Strictly stable distributions on convex cones. Electron. J. Prob. 13, 259-321. · Zbl 1196.60028 · doi:10.1214/EJP.v13-487 [7] de Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in \(C[0,1]\). Ann. Prob. 29, 467-483. · Zbl 1010.62016 · doi:10.1214/aop/1008956340 [8] Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249-274. · Zbl 1070.60046 · doi:10.1016/j.spa.2004.09.003 [9] Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N. S.) 80, 121-140. · Zbl 1164.28005 · doi:10.2298/PIM0694121H [10] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principles Math. Sci. 288 ), 2nd edn. Springer, Berlin. · Zbl 1018.60002 [11] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd edn. Springer, New York. · Zbl 0996.60001 [12] Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671-699. · Zbl 1090.90017 · doi:10.1239/jap/1037816012 [13] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66-138. · Zbl 0597.60048 · doi:10.2307/1427239 [14] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance . Chapman & Hall, New York. · Zbl 0925.60027 [15] 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. · Zbl 0601.60085 · doi:10.1007/978-1-4615-8162-8_3 [16] Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67-85. · Zbl 0428.60010 · doi:10.1287/moor.5.1.67 [17] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues . Springer, New York. · Zbl 0993.60001 · doi:10.1007/b97479 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.