Structural, continuity, and asymptotic properties of a branching particle system

Lithuanian Mathematical Journal

Volumn 49, Issue 3, 2009, Pages 241-270

  Hochberg, K J ^a   Vinogradov, V ^b  

^a BAR ILAN UNIVERSITY   (Israel)

^b OHIO UNIVERSITY   (United States)

Author keywords

Asymptotic expansion; Athreya Ney type representation; Babel class; Bessel function; Birth death Markov process; Branching diffusing particle system; Cluster structure; Continuity of stochastic process in mean square; Dawson Watanabe superprocess; Esscher transform; J rgensen type convergence; Laplace method; Leading error term; Levy process; Local limit theorem; P lya Aeppli law; Poisson mixture; Poisson exponential law; Power variance family; Summation invariance; Unit variance function; Yaglom type exponential limit law; Zero modified geometric law

Indexed keywords

EID: 71349087408     PISSN: 03631672     EISSN: 15738825     Source Type: Journal    
DOI: 10.1007/s10986-009-9049-5     Document Type: Article

References (55)

1. A. Aleškevičienė and V. Statulevičius, Theorems of large deviations in the approximation by the compound Poisson distribution, Acta Appl. Math., 78: 21-34, 2003.
2. K. B. Athreya and P. E. Ney, Branching Processes, Springer, New York, 1972.
3. A. D. Barbour, O. Chryssaphinou, and E. Vaggelatou, Applications of compound Poisson approximation, in C. A. Charalambides, M. V. Koutras, and N. Balakrishnan (Eds.), Probability and Statistical Models with Applications, Chapman & Hall, Boca-Raton, 2001, pp. 41-62.
4. M. Baron and A. L. Rukhin, Distribution of the number of visits of a random walk, Commun. Stat. Stochast. Models, 15: 593-597, 1999.
5. M. V. Boutsikas, Compound Poisson process approximation for locally dependent real-valued random variables via a new coupling equality, Bernoulli, 12: 501-514, 2006.
6. M. V. Boutsikas and M. V. Koutras, Modeling claim exceedances over thresholds, Insur. Math. Econ., 30: 67-83, 2002.
7. M. V. Boutsikas and M. V. Koutras, On the number of overflown urns and excess balls in an allocation model with limited urn capacity, J. Stat. Plann. Infer., 104: 259-286, 2002.
8. D. R. Cox, Renewal Theory, Methuen, London, 1962.
9. H. Cramér and M. R. Leadbetter, Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications, Wiley, New York, 1967.
10. D. A. Dawson and K. Fleischmann, Strong clumping of critical space-time branching models in subcritical dimensions, Stoch. Process. Appl., 30: 193-208, 1988.
11. D. A. Dawson, K. J. Hochberg, and V. Vinogradov, On path properties of super-2 processes I, in D. A. Dawson (Ed.), Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Systems, CRM Proc. Lecture Notes, Vol. 5, Amer. Math. Soc., Providence, 1994, pp. 69-82.
12. D. A. Dawson, K. J. Hochberg, and V. Vinogradov, On path properties of super-2 processes II, in M. G. Cranston and M. A. Pinsky (Eds.), Stochastic Analysis, Proc. Sympos. Pure Math., Vol. 57, Amer. Math. Soc., Providence, 1995, pp. 383-403.
13. D. A. Dawson, K. J. Hochberg, and V. Vinogradov, High-density limits of hierarchically structured branching-diffusing populations, Stoch. Process. Appl., 62: 191-222, 1996.
14. D. A. Dawson, K. J. Hochberg, and V. Vinogradov, On weak convergence of branching particle systems undergoing spatial motion, in J.-P. Fouque, K. J. Hochberg, and E. Merzbach (Eds.), Stochastic Analysis: Random Fields and Measure-Valued Processes, Israel Math. Conf. Proc., Vol. 10, Bar-Ilan Univ., Ramat-Gan, 1996, pp. 65-79.
15. D. A. Dawson and V. Vinogradov, Almost-sure path properties of (2, d, β)-superprocesses, Stoch. Process. Appl., 51: 221-258, 1994.
16. R. R. Delongchamp, C. R. Valentine, and H. V. Malling, Estimation of the average burst size of ΦX174 am3, cs70 for use in mutation assays with transgenic mice, Environ. Mol. Mutagen., 37: 356-360, 2001.
17. E. B. Dynkin, An Introduction to Branching Measure-Valued Processes, Amer. Math. Soc., Providence, 1994.
18. M. V. Fedoryuk, Asymptotics: Integrals and Series, Nauka, Moscow, 1987 (in Russian).
19. M. V. Fedoryuk, Asymptotic methods in analysis, in R. V. Gamkrelidze (Ed.), Analysis I. Integral Representations and Asymptotic Methods, Vol. 1, part 2, Springer, Berlin, 1989, pp. 83-191.
20. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, 1954.
21. L. G. Gorostiza, Space scaling limit theorems for infinite particle branching Brownian motions with immigration, in M. Metivier and É. Pardoux (Eds.), Stochastic Differential Systems. Filtering and Control. Proc. 4th IFIP-WG 7/1 Working Conf. held in Marseille-Luminy, March 12-17, 1984, Lecture Notes in Control and Inform. Sci., Vol. 69, Springer, Berlin, 1985, pp. 91-99.
22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition, Academic, Oxford, 2007.
23. T. E. Harris, The Theory of Branching Processes, Springer, Berlin, 1963.
24. P. Hougaard, M.-L. T. Lee, and G. A. Whitmore, Analysis of overdispersed count data by mixtures of Poisson variables and Poisson processes, Biometrics, 53: 1225-1238, 1997.
25. N. L. Johnson, A. W. Kemp, and S. Kotz, Univariate Discrete Distributions, 3rd edition, Wiley, Hoboken, NJ, 2005.
26. B. Jørgensen, Some properties of exponential dispersion models, Scand. J. Stat., 13: 187-197, 1986.
27. B. Jørgensen, Exponential dispersion models and extensions: A review, Int. Stat. Rev., 60: 5-20, 1992.
28. B. Jørgensen, The Theory of Dispersion Models, Chapman & Hall, London, 1997.
29. B. Jørgensen, J. R. Martínez, and M. Tsao, Asymptotic behaviour of the variance function, Scand. J. Stat., 21: 223-243, 1994.
30. B. Jørgensen, J. R. Martínez, and V. Vinogradov, Domains of attraction to Tweedie distributions, Lith. Math. J., 49(4), 2009.
31. B. Jørgensen and M. C. P. de Souza, Fitting Tweedie's compound Poisson model to insurance claims data, Scand. Actuar. J., 1: 69-93, 1994.
32. B. Jørgensen and V. Vinogradov, Convergence to Tweedie models and related topics, in N. Balakrishnan (Ed.), Advances on Theoretical and Methodological Aspects of Probability and Statistics, Vol. 2, Chap. 30, Francis & Taylor, London, 2002, pp. 473-488.
33. W. S. Kendal, Spatial aggregation of the Colorado potato beetle described by an exponential dispersion model, Ecol. Model., 151: 261-269, 2002.
34. W. S. Kendal, A scale invariant clustering of genes on human chromosome 7, BMC Evol. Biol., 4(3), 2004.
35. W. S. Kendal, Scale invariant correlations between genes and snps on human chromosome 1 reveal potential evolutionary mechanisms, J. Theor. Biol., 245: 329-340, 2007.
36. W. S. Kendal, F. J. Lagerwaard, and O. Agboola, Characterization of the frequency distribution for human hematogenous metastases: Evidence for clustering and a power variance function, Clin. Exp. Metastas., 18: 219-229, 2000.
37. F. B. Knight, Essentials of Brownian Motion and Diffusion, Amer. Math. Soc., Providence, 1981.
38. C. C. Kokonendji, S. Dossou-Gbété, and C. G. B. Demétrio, Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demetrio classes, SORT, 28: 201-214, 2004.
39. A. N. Kolmogorov and N. A. Dmitriev, Branching random processes, Dokl. Akad. Nauk SSSR, 56(1): 7-10, 1947 (in Russian). English translation in A. N. Shiryayev (Ed.), SelectedWorks of A. N. Kolmogorov, Vol. 2, Kluwer, Dordrecht, 1992, pp. 309-314.
40. U. Kuchler and M. Sorensen, Exponential Families of Stochastic Processes, Springer, New York, 1997.
41. J.-F. Le Gall, Spatial Branching Processes, Random Snakes, and Partial Differential Equations, Birkhäuser, Basel, 1999.
42. G. Letac, Lectures on Natural Exponential Families and Their Variance Functions, Monogr. Math., Vol. 50, Instituto de Mathematica Pura e Aplicada, Rio de Janeiro, 1992.
43. J. Neveu and J. W. Pitman, The branching process in a Brownian excursion, in Lecture Notes in Math., Vol. 1372, Springer, Berlin, 1989, pp. 248-257.
44. H. H. Panjer and G. E. Willmot, Insurance Risk Models, Society of Actuaries, Schaumburg IL, 1992.
45. E. A. Perkins, Dawson-Watanabe superprocesses and measure-valued diffusions, in P. Bernard (Ed.), Lectures on Probability Theory and Statistics/Ecole d'Eté de Probabilités de Saint Flour XXIX - 1999, Lecture Notes in Math., Vol. 1781, Springer, Berlin, 2002, pp. 125-329.
46. S. Robin, A compound Poisson model for word occurrences in DNA sequences, Appl. Stat., 51: 437-451, 2002.
47. B. A. Sevast'yanov, Transient phenomena in branching stochastic processes, Theory Probab. Appl., 4: 113-128, 1959.
48. G. K. Smyth and B. Jørgensen, Fitting Tweedie's compound Poisson model to insurance claims data: Dispersion modelling, Astin Bull., 32: 143-157, 2002.
49. O. E. Trofimov and D. G. Frizen, Coefficients of the asymptotic expansion of integrals via Laplace method, Automat. Monitoring Measuring, 2: 97-98, 1981.
50. V. Vinogradov, On the power-variance family of probability distributions, Commun. Stat. Theory Methods, 33: 1007-1029, 2004 (Errata p. 2573).
51. V. Vinogradov, Local approximations for branching particle systems, Commun. Stoch. Anal., 1: 293-309, 2007.
52. V. Vinogradov, On infinitely divisible exponential dispersion model related to Poisson-exponential distribution, Commun. Stat. Theory Methods, 36: 253-263, 2007.
53. V. Vinogradov, On structural and asymptotic properties of some classes of distributions, Acta Appl. Math., 97: 335-351, 2007.
54. V. Vinogradov, Properties of certain Lévy and geometric Lévy processes, Commun. Stoch. Anal., 2: 193-208, 2008.
55. X. Zhou, A zero-one law of almost sure local extinction for (1 + β)-super-Brownian motion, Stoch. Process. Appl., 118: 1982-1996, 2008.