Weak and Strong Convergence of the Distributions of Counting Processes
Yu. M. Kabanov, R. Liptser, Albert Nikolaevich Shiryaev · Theory of Probability and Its Applications · 1984
Previous article Next article Weak and Strong Convergence of the Distributions of Counting ProcessesYu. M. Kabanov, R. Sh. Liptser, and A. N. ShiryaevYu. M. Kabanov, R. Sh. Liptser, and A. N. Shiryaevhttps://doi.org/10.1137/1128026PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. Sh. Liptser and , A. N. Shiryaev, Weak convergence of semimartingales to stochastically continuous processes with independent and conditionally independent increments, Mat. Sb. (N.S.), 116(158) (1981), 331–358, 463, (In Russian.) 84b:60041 0484.60024 Google Scholar[2] J. Jacod, , A. Kłopotowski and , J. Mémin, Théorème de la limite centrale et convergence fonctionnelle vers un processus à accroissements indépendants: la méthode des martingales, Ann. Inst. H. Poincaré Sect. B (N.S.), 18 (1982), 1–45 83f:60046 0493.60033 Google Scholar[3] B. Grigelionis and , R. Mikulavichus, On weak convergence of semimartingales and point processesStochastic differential systems (Visegrád, 1980), Lecture Notes in Control and Information Sci., Vol. 36, Springer, Berlin, 1981, 61–68 83g:60057 0487.60041 CrossrefGoogle Scholar[4] B. Grigelionis and , R. Mikuliavichus, On the weak convergence of semimartingales, Litovsk. Mat. Sb., 21 (1981), 9–24, (In Russian.) 83c:60073 Google Scholar[5] Yu. M. Kabanov, , R. Sh. Liptser and , A. N. Shiryaev, Martingale methods in the theory of point processes, Trudy Shkoly-Seminars po Theorii Sluchainykh Protsessov (Druskininkai, 1974), Vol. 2, Inst. Fiziki i Matemat. Akad. Nauk Litov. SSR, Wilno, 1975, 269–354, (In Russian.) Google Scholar[6] R. S. Liptser and , A. N. Shiryayev, Statistics of Random Processes, Vol. II, Springer-Verlag, Berlin, 1978 CrossrefGoogle Scholar[7] Jean Jacod, Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 235–253 52:1875 CrossrefGoogle Scholar[8] B. Grigelionis, Characterization of random processes with conditionally independent increments, Litovsk. Mat. Sb., 15 (1975), 53–60, 241, (In Russian.) 54:8886 Google Scholar[9] Yu. M. Kabanov, , R. Sh. Liptser and , A. N. Shiryaev, Some limit theorems for simple point processes (a martingale approach), Stochastics, 3 (1980), 203–216 82a:60027 0441.60045 CrossrefGoogle Scholar[10] T. Brcwn, A martingale approach to the Poisson convergence of simple point processes, Ann. Probab., 6 (1978), 615–628 58:3023 CrossrefGoogle Scholar[11] T. Brown, Compensators and Cox convergence, Math. Proc. Cambridge Philos. Soc., 90 (1981), 305–319 82i:60094 0474.60041 CrossrefGoogle Scholar[12] J. Jacod and , J. Mémin, Sur la convergence des semimartingales vers un processus à accroissements indépendantsSeminar on Probability, XIV (Paris, 1978/1979) (French), Lecture Notes in Math., Vol. 784, Springer, Berlin, 1980, 227–248 83i:60060 0433.60034 CrossrefGoogle Scholar[13] C. R. Rao, Linear statistical inference and its applications, John Wiley & Sons, New York-London-Sydney, 1973xx+625 49:11677 0256.62002 CrossrefGoogle Scholar[14] D. L. McLeish, An extended martingale invariance principle, Ann. Probability, 6 (1978), 144–150 57:10774 0379.60046 CrossrefGoogle Scholar[15] P. Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[16] C. Stone, Weak convergence of stochastic processes defined on semi-infinite time intervals, Proc. Amer. Math. Soc., 14 (1963), 694–696 27:3015 0116.35602 CrossrefGoogle Scholar[17] J. Jacod and , J. Mémin, Un nouveau critère de compacité relative pour une suite de processusSeminar on Probability, Rennes 1979 (French), Univ. Rennes, Rennes, 1979, Exp. No. 4, 27 82k:60110 Google Scholar[18] David Aldous, Stopping times and tightness, Ann. Probability, 6 (1978), 335–340 57:14086 0391.60007 CrossrefGoogle Scholar[19] R. Rebolledo, La méthode des martingales appliquée à l'étude de la convergence en loi de processus, Bull. Soc. Math. France Mém., (1979), v+125 pp. (1980) 81g:60002 0425.60036 Google Scholar[20] E. Lenglart, Relation de domination entre deux processus, Ann. Inst. H. Poincaré Sect. B (N.S.), 13 (1977), 171–179 57:10810 0373.60054 Google Scholar[21] B. Grigelionis and , R. Mikuliavichus, On the weak convergence of random point processes, Litovsk. Mat. Sb., 21 (1981), 49–55, (In Russian.) 83g:60012 0491.60033 Google Scholar[22] T. Brown, Poisson approximations and exchangeable random variablesExchangeability in probability and statistics (Rome, 1981), North-Holland, Amsterdam, 1982, 177–183 84e:60036 0493.60027 Google Scholar[23] E. Valkeila, A general Poisson approximation theorem, Stochastics, 7 (1982), 159–171 83m:60044 0491.60022 CrossrefGoogle Scholar[24] Yu. M. Kabanov, On the rate of convergence of distributions of counting processes to the distribution of a counting process with independent increments, Soviet Math. Dokl., 25 (1982), 794–797 Google Scholar[25] R. J. Serfling, Some elementary results on Poisson approximation in a sequence of Bernoulli trials, SIAM Rev., 20 (1978), 567–579 10.1137/1020070 58:2993 0383.60027 LinkGoogle Scholar[26] L. Le Cam, An approximation theorem for the Poisson binomial distribution, Pacific J. Math., 10 (1960), 1181–1197 25:5567 0118.33601 CrossrefGoogle Scholar[27] Yu. V. Prokhorov, Asymptotic behavior of the binomial distribution, Uspehi Matem. Nauk (N.S.), 8 (1953), 135–142, (In Russian.) 15,138g Google Scholar[28] L. LeCam, J. Neyman and , L. LeCam, On the distributions of sums of independent random variablesBernoulli, Bayes, Laplace, Springer-Verlag, Berlin, 1965 CrossrefGoogle Scholar[29] S. Ya. Shorgin, Approximation of a generalized binomial distribution, Theory Prob. Appl., 22 (1977), 846–850 LinkGoogle Scholar[30] Yu. M. Kabanov and , R. Sh. Liptser, On convergence in variation of the distributions of multivariate point processes, Z. Wahrsch. Verw. Gebiete, 63 (1983), 475–485 85f:60071 0532.60042 CrossrefGoogle Scholar[31] Yu. M. Kabanov, , R. Sh. Liptser and , A. N. Shiryaev, Absolute continuity and singularity of locally absolutely continuous probability distributions. I, Mat. Sb. (N.S.), 107(149) (1978), 364–415, 463 80e:60056a Yu. M. Kabanov, , R. Sh. Liptser and , A. N. Shiryaev, Absolute continuity and singularity of locally absolutely continuous probability distributions. II, Mat. Sb. (N.S.), 108(150) (1979), 32–61, 143, (In Russian.) 80e:60056b Google Scholar[32] C. Dellachérie, Capacités et processus stochastiques, Springer-Verlag, Berlin, 1972ix+155, New York 56:6810 0246.60032 CrossrefGoogle Scholar[33] P. A. Meyer, Probability and potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966xiii+266 34:5119 0138.10401 Google Scholar[34] R. Sh. Liptser and , A. N. Shiryaev, Weak convergence of a sequence of semimartingales to a process of diffusion type, Mat. Sb. (N.S.), 121(163) (1983), 176–200 84j:60012 0521.60034 Google Scholar[35] J. Mémin, Distance en variation et conditions de contiquité pour des lois de processes ponctuels, 1982, preprint, Univ. de Rennes, Rennes Google Scholar[36] B. A. Sevastyanov, Branching Processes, Nauka, Moscow, 1971, (In Russian.) Google Scholar[37] V. V. Anisimov and , V. N. Sityuk, Asymptotic behavior of an inhomogeneous Poisson process with a leading function depending on a small parameter controlled by a Markov chain, Cybernetics, 4 (1977), 149–150, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Poisson approximationProbability Surveys, Vol. 16, No. none Cross Ref Separation of time-scales and model reduction for stochastic reaction networksThe Annals of Applied Probability, Vol. 23, No. 2 Cross Ref A Poisson Limit Theorem for Reliability Models Based on Markov ChainsCommunications in Statistics - Theory and Methods, Vol. 35, No. 1 Cross Ref Poisson approximation for some point processes in reliability1 July 2016 | Advances in Applied Probability, Vol. 36, No. 2 Cross Ref Poisson approximation, compensators and couplingStochastic Analysis and Applications, Vol. 18, No. 1 Cross Ref On mixed poisson processes and martingalesScandinavian Actuarial Journal, Vol. 1998, No. 1 Cross Ref A first order approximation forthe convergence of distributionsof the cox processes with4 April 2007 | Stochastics and Stochastic Reports, Vol. 54, No. 3-4 Cross Ref A note on the prohorov distance between a counting process and a poisson process4 April 2007 | Stochastics and Stochastic Reports, Vol. 45, No. 1-2 Cross Ref On the optimality of multivariate Poisson approximationStochastic Processes and their Applications, Vol. 44, No. 1 Cross Ref On the Weak Convergence of Counting Processes with Nuisance ParametersV. I. Pagurova and S. A. Nesterova17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 1AbstractPDF (734 KB)Weak convergence of jump processes29 September 2006 Cross Ref A prohorov bound for a poisson process and an arbitrary counting process with some applications4 April 2007 | Stochastics and Stochastic Reports, Vol. 37, No. 3 Cross Ref Chapter 1 Point processes Cross Ref Poisson approximations of multinomial distributions and point processesJournal of Multivariate Analysis, Vol. 25, No. 1 Cross Ref On the Problem of the Distributions of U-StatisticsA. A. Grusho17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 2AbstractPDF (477 KB)On the Levy-Prohorov distance between counting processesStochastic Processes and their Applications, Vol. 26 Cross Ref On Convergence of Counting Processes Associated with U-StatisticsA. A. Grusho17 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 3AbstractPDF (376 KB)An Estimate of Closeness in Variation of Probability MeasuresYu. M. Kabanov17 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 2AbstractPDF (480 KB)Partitions of point processes: Multivariate poisson approximationsStochastic Processes and their Applications, Vol. 20, No. 2 Cross Ref A Note on One-Dimensional Distances between Two Counting ProcessesM. Nikunen and E. Valkeila17 July 2006 | Theory of Probability & Its Applications, Vol. 29, No. 3AbstractPDF (349 KB)Software Reliability Modeling Cross Ref Volume 28, Issue 2| 1984Theory of Probability & Its Applications History Submitted:01 December 1982Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1128026Article page range:pp. 303-336ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics