Skorokhod’s M1 topology for distribution-valued processes

Electronic Communications in Probability

Volumn 21, Issue , 2016, Pages

  Ledger, Sean ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Compacntess and tightness characterisation; Countably Hilbertian nuclear space; Skorokhod M1 topology; Tempered distribution

Indexed keywords

EID: 84965042590     PISSN: None     EISSN: 1083589X     Source Type: Journal    
DOI: 10.1214/16-ECP4754     Document Type: Article

References (35)

1. F. Avram and M.S. Taqqu. Probability bounds for M-Skorohod oscillations. Stochastic Processes and their Applications, 33(1):63-72, 1989. MR-1027108
2. F. Avram and M.S. Taqqu. Weak convergence of sums of moving averages in the α-stable domain of attraction. The Annals of Probability, 20(1):483-503, 1992. MR-1143432
3. B. Basrak and D. Krizmanić. A Multivariate Functional Limit Theorem in Weak M1 Topology. Journal of Theoretical Probability, 28:119-136, 2015. MR-3320962
4. J.J. Becnel and A.N. Sengupta. Nuclear space facts, strange and plain. http://www.math.lsu.ed/~sengupta/papers/BecSenNuclear.pdf, 2009.
5. G. Ben Arous and J. Černy. Scaling limit for trap models on Zd. The Annals of Probability, 35(6):2356-2384, 2007. MR-2353391
6. P. Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley-Interscience Publication. MR-1700749
7. D. Blount and M.A. Kouritzin. On convergence determining and separating classes of functions. Stochastic Processes and their Applications, 120:1898-1907, 2010. MR-2673979
8. N. Bush, B.M. Hambly, H. Haworth, L. Jin, and C. Reisinger. Stochastic Evolution Equations in Portfolio Credit Modelling. SIAM Journal of Financial Mathematics, 2(1):627-664, 2011. MR-2836495
9. F. Delarue, J. Inglis, S. Rubenthaler, and E. Tanré. Particle systems with singular mean-field self-excitation. Application to neuronal networks. Stochastic Processes and their Applications, 125(6):2451-2492, 2015. MR-3322871
10. K. R. Duffy, C. Macci, and G.L. Torrisi. Sample path large deviations for order statistics. Journal of Applied Probability, 48(1):238-257, 2011. MR-2809898
11. S. Ethier and T. Kurtz. Markov Processes: Characterization and Convergence. Wiley, 1986. MR-0838085
12. J.M. Harrison and R.J. William. A multiclasss closed queueing network with unconventional heavy traffic behavior. The Annals of Applied Probability, 6(1):1-47, 1996. MR-1389830
13. R. Holley and D.W. Stroock. Central limit phenomena of various interacting systems. The Annals of Mathematics, 110(2):333-393, 1979. MR-0549491
14. K. Itô. Distribution-valued processes arising from independent Brownian motions. Mathematische Zeitschrift, 182(1):17-33, 1983. MR-0686883
15. S. Iyengar. Hitting lines with two-dimensional Brownian motion. SIAM Journal of Applied Mathematics, 45(6):983-989, 1985. MR-0813460
16. A. Jakubowski. On the Skorokhod topology. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 22(3):263-285, 1986. MR-0871083
17. G. Kallianpur and J. Xiong. Stochastic Differential Equations in Infinite Dimensional Spaces. Institute of Mathematical Statistics: Lecture notes, monograph series. Institute of Mathematical Statistics, 1995. MR-1465436
18. O. Kella and W. Whitt. Diffusion Approximations for Queues of Server Vacations. Advances in Applied Probability, 22(3):706-729, 1990. MR-1066971
19. M.A. Kouritzin. On tightness of probability measures on Skorokhod spaces. http://hdl. handle.net/10402/era.37299, 2014.
20. D. Krizmanić. Weak convergence of partial maxima processes in the M1 topology. Extremes, 17(3):447-465, 2014. MR-3252821
21. A. Mandelbaum and W.A. Massey. Strong approximations for time-dependent queues. Mathematics of Operations Research, 20(1):33-64, 1995. MR-1320446
22. A. Metzler. On the first passage problem for correlated Brownian motion. Statistics & Probability Letters, 80(5-6):277-284, 2010. MR-2593563
23. I. Mitoma. On the sample continuity of S0-processes. Journal of the Mathematical Society of Japan, 35(4):629-636, 1983. MR-0714465
24. I. Mitoma. Tightness of probabilities on C ([0; 1]; S0) and D([0; 1]; S0). The Annals of Probability, 11(4):989-999, 1983. MR-0714961
25. G. Pang and W. Whitt. Continuity of a queueing integral representation in the M1 topology. The Annals of Applied Probability, 20(1):214-237, 2010. MR-2582647
26. I. Pavlyukevich and M. Riedle. Non-standard skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces. Stochastic Analysis and Applications, 33:271-305, 2015. MR-3305470
27. S. Resnick and E. van der Berg. Weak convergence of high-speed network traffic models. Journal of Applied Probability, 37:575-597, 2000. MR-1781014
28. A.V. Skorokhod. Limit Theorems for Stochastic Processes. Theory of Probability and its Applications, 1(3):261-290, 1956. MR-0084897
29. O.G. Smolyanov and S.V. Fomin. Measures on linear topological spaces. Russian Mathematical Surveys, 31(4):1-53, 1976.
30. A.-S. Sznitman. Topics in propagation of chaos. In Paul-Louis Hennequin, editor, Ecole d’Eté de Probabilités de Saint-Flour XIX-1989, volume 1464 of Lecture Notes in Mathematics, chapter 3, pages 165-251. Springer Berlin Heidelberg, 1991. MR-1108185
31. W. Whitt. Weak Convergence of First Passage Time Processes. Journal of Applied Probability, 8(2):417-422, 1971. MR-0307335
32. W. Whitt. Limits for cumulative input processes to queues. Probability in Engineering and Informational Sciences, 14:123-150, 2000. MR-1757175
33. W. Whitt. The Reflection Map with Discontinuities. Mathematics of Operations Research, 26(3):447-484, 2001. MR-1849880
34. W. Whitt. Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research and Financial Engineering. Springer, 2002. MR-1876437
35. M.J. Wichura. Functional laws of the iterated logarithm for the partial sums of i.i.d. random variables in the domain of attraction of a completely asymmetric stable law. The Annals of Probability, 2(6):1108-1138, 1974. MR-0358950