Abstract:
The theme of the article is the convergence of distributions of counting processes. The paper contains several theorems connecting the convergence of predictable characteristics (compensators) with the convergence of distributions. If the limit process has independent (or conditionally independent) increments, we use the method of «strochastic exponentials»; by means of this method we obtain an estimate of the rate of convergence
of finite-dimensional distributions to the corresponding distributions of the Poisson process. Techniques based on the compactness criterion in used to prove a weak convergence to a counting process with a (random) continuous compensator. We present also a criterion for the convergence in variation together with the estimates of the rate of convergence. As an illustration we investigate the strong convergence of conditionally Poisson processes with intensities depending on a Markov process. Another example is an estimate of the rate of convergence of counting processes connected with the empirical distribution functions to the Poisson process.
Citation:
Yu. M. Kabanov, R. Š. Lipcer, A. N. Širyaev, “Weak and strong convergence of distributions of counting processes”, Teor. Veroyatnost. i Primenen., 28:2 (1983), 288–319; Theory Probab. Appl., 28:2 (1984), 303–336
\Bibitem{KabLipShi83}
\by Yu.~M.~Kabanov, R.~{\v S}.~Lipcer, A.~N.~{\v S}iryaev
\paper Weak and strong convergence of distributions of counting processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1983
\vol 28
\issue 2
\pages 288--319
\mathnet{http://mi.mathnet.ru/tvp2296}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=700211}
\zmath{https://zbmath.org/?q=an:0533.60055|0516.60056}
\transl
\jour Theory Probab. Appl.
\yr 1984
\vol 28
\issue 2
\pages 303--336
\crossref{https://doi.org/10.1137/1128026}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1984SS85900006}
Linking options:
https://www.mathnet.ru/eng/tvp2296
https://www.mathnet.ru/eng/tvp/v28/i2/p288
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S. Y. Novak, “Poisson approximation”, Probab. Surveys, 16:none (2019)
Hye-Won Kang, Thomas G. Kurtz, “Separation of time-scales and model reduction for stochastic reaction networks”, Ann. Appl. Probab., 23:2 (2013)
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James Ledoux, Handbook of Reliability Engineering, 2003, 213
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Bronius Grigelionis, “On mixed poisson processes and martingales”, Scandinavian Actuarial Journal, 1998:1 (1998), 81
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Hans-Jürgen Witte, “On the optimality of multivariate Poisson approximation”, Stochastic Processes and their Applications, 44:1 (1993), 75
Xia Aihua, “A note on the prohorov distance between a counting process and a poisson process”, Stochastics and Stochastic Reports, 45:1-2 (1993), 61
Aihua Xia, Lecture Notes in Mathematics, 1526, Séminaire de Probabilités XXVI, 1992, 32
Martti Nikunen, Esko Valkeila, “A prohorov bound for a poisson process and an arbitrary counting process with some applications”, Stochastics and Stochastic Reports, 37:3 (1991), 133
V. I. Pagurova, S. A. Nesterova, “Weak convergence of counting processes in the presence of nuisance parameters”, Theory Probab. Appl., 36:1 (1991), 176–185
Richard F. Serfozo, Handbooks in Operations Research and Management Science, 2, Stochastic Models, 1990, 1
Paul Deheuvels, Dietmar Pfeifer, “Poisson approximations of multinomial distributions and point processes”, Journal of Multivariate Analysis, 25:1 (1988), 65
Martti Nikunen, Esko Valkeila, “On the Levy-Prohorov distance between counting processes”, Stochastic Processes and their Applications, 26 (1987), 190
A. A. Grusho, “On Distributions of U-Statistics”, Theory Probab. Appl., 32:2 (1987), 369–373
Richard F Serfozo, “Partitions of point processes: Multivariate poisson approximations”, Stochastic Processes and their Applications, 20:2 (1985), 281
Yu. M. Kabanov, “An estimate of the variation distance between probability measures”, Theory Probab. Appl., 30:2 (1986), 413–417
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