A functional limit theorem without centering for general shot noise processes
Abstract
UDC 519.27
We define a general shot noise process as the convolution of a deterministic càdlàg function and a locally finite counting process concentrated on the nonnegative halfline. In this paper, we provide the sufficient conditions ensuring that a general shot noise process properly normalized without centering converges weakly in the Skorokhod space. We give several examples of particular counting processes satisfying the sufficient conditions and formulate the corresponding limit theorems. The present work continues the investigation initiated in [Iksanov and Rashytov (2020)], where a functional limit theorem with centering was proved under the condition that the limit process is a Riemann–Liouville-type (Gaussian) process.
References
P. Billingsley, Convergence of probability measures, 2nd ed., John Wiley and Sons, New York (1999), https://doi.org/10.1002/9780470316962 DOI: https://doi.org/10.1002/9780470316962
N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Cambridge Univ. Press (1989).
C. Dong, A. Iksanov, Weak convergence of random processes with immigration at random times, J. Appl. Probab., 57 , № 1, 250 – 265 (2020), https://doi.org/10.1017/jpr.2019.88 DOI: https://doi.org/10.1017/jpr.2019.88
A. Gut, Stopped random walks. Limit theorems and applications, 2nd ed., Springer Series in Operations Research and Financial Engineering. Springer, New York, (2009), https://doi.org/10.1007/978-0-387-87835-5 DOI: https://doi.org/10.1007/978-0-387-87835-5
A. Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions, Stoch.Process and Appl., 123 , № 6, 1987 – 2010 (2013), https://doi.org/10.1016/j.spa.2013.01.019 DOI: https://doi.org/10.1016/j.spa.2013.01.019
A. Iksanov, Z. Kabluchko, A. Marynych, G. Shevchenko, Fractionally integrated inverse stable subordinators, Stoch. Process and Appl., 127, № 1, 80 – 106 (2016), https://doi.org/10.1016/j.spa.2016.06.001 DOI: https://doi.org/10.1016/j.spa.2016.06.001
A. Iksanov, M. Meiners, Exponential rate of almost sure convergence of intrinsic martingales in supercritical branching random walks, J. Appl. Probab., 47 , №2, 513 – 525 (2010), https://doi.org/10.1239/jap/1276784906 DOI: https://doi.org/10.1239/jap/1276784906
A. Iksanov, B. Rashytov, A functional limit theorem for the general shot noise processes, J. Appl. Probab., 57 , №1, 280 – 294 (2020), https://doi.org/10.1017/jpr.2019.95 DOI: https://doi.org/10.1017/jpr.2019.95
J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, (2003), https://doi.org/10.1007/978-3-662-05265-5 DOI: https://doi.org/10.1007/978-3-662-05265-5
M. M. Meerschaert, H. P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab., 41 , № 3, 623 – 638 (2004), https://doi.org/10.1239/jap/1091543414 DOI: https://doi.org/10.1239/jap/1091543414
T. Owada, G. Samorodnitsky, Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows, Ann. Probab., 43 , № 1, 240 – 285 (2015), https://doi.org/10.1214/13-AOP899 DOI: https://doi.org/10.1214/13-AOP899
G. Pang, Y. Zhou, Functional limit theorems for shot noise processes with weakly dependent noises, Stoch. Syst., 10 , № 2, 99 – 123 (2020), https://doi.org/10.1287/stsy.2019.0051 DOI: https://doi.org/10.1287/stsy.2019.0051
S. I. Resnick, Heavy-tail phenomena: probabilistic and statistical modeling, Springer Series in Operations Research and Financial Engineering. Springer, New York, (2007).
S. Resnick, P. Greenwood, A bivariate stable characterization and domains of attraction, J. Multivar. Anal., 9 , № 2, 206 – 221 (1979), https://doi.org/10.1016/0047-259X(79)90079-4 DOI: https://doi.org/10.1016/0047-259X(79)90079-4
M. Tamborrino, P. Lansky, Shot noise, weak convergence and diffusion approximations, Phys. D 418 (2021), 132845. https://arxiv.org/abs/2005.06067, https://doi.org/10.1016/j.physd.2021.132845 DOI: https://doi.org/10.1016/j.physd.2021.132845
M. Yamazato, On a J1 -convergence theorem for stochastic processes on $D[0,infty )$ having monotone sample paths and its applications, RIMS Kˆokyˆuroku, 1620 , 109 – 118 (2009).
G. Ver`ovkin, O. Marinich, Staczionarni graniczi proczesiv drobovogo efektu, Teoriya Imovirnoste ta Matematychna Statystyka, 101, 63 – 77 (2019).
Copyright (c) 2021 Alexander Iksanov
This work is licensed under a Creative Commons Attribution 4.0 International License.
