Geodesic
Limit distributions for doubly stochastically rarefied renewal processes and their properties
Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 4, pp. 753-773 Cet article a éte moissonné depuis la source Math-Net.Ru

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A limit theorem is proved for doubly stochastically rarefied renewal processes. It is shown that under rather general conditions, as limit laws in limit theorems for mixed geometric random sums, there appear mixed exponential and mixed Laplace distributions. Some known and new properties of these distributions are reviewed. Also, some nonobvious properties of special representatives of these classes (the Weibull, Mittag-Leffler, Linnik, and other distributions) are described.
Keywords: doubly stochastically rarefied renewal process, mixed geometric distribution, mixed geometric random sum, mixed exponential distribution, Weibull distribution, Linnik distribution
Mots-clés : stable distribution, Mittag-Leffler distribution, Laplace distribution. 
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V. Yu. Korolev. Limit distributions for doubly stochastically rarefied renewal processes and their properties. Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 4, pp. 753-773. http://geodesic.mathdoc.fr/item/TVP_2016_61_4_a6/

[1] Zolotarev V. M., Odnomernye ustoichivye raspredeleniya, Nauka, M., 1983, 304 pp.

[2] Kovalenko I. N., “O klasse predelnykh raspredelenii dlya redeyuschikh potokov odnorodnykh sobytii”, Lit. matem. sb., 5:4 (1965), 569–573 | Zbl

[3] Korolev V. Yu., “Skhodimost sluchainykh posledovatelnostei s nezavisimymi sluchainymi indeksami. I”, Teoriya veroyatn. i ee primen., 39:2 (1994), 313–333 | Zbl

[4] Korolev V. Yu., “Skhodimost sluchainykh posledovatelnostei s nezavisimymi sluchainymi indeksami. II”, Teoriya veroyatn. i ee primen, 40:4 (1995), 907–910 | Zbl

[5] Korolev V. Yu., Bening V. E., Shorgin S. Ya., Matematicheskie osnovy teorii riska, 2, FIZMATLIT, M., 2011, 620 pp.

[6] Korolev V. Yu., Bening V. E., Zaks L. M., Zeifman A. I., “Obobschennoe raspredelenie Laplasa kak predelnoe dlya sluchainykh summ i statistik, postroennykh po vyborkam sluchainogo ob'ema”, Informatika i ee primen., 6:4 (2012), 34–39

[7] Linnik Yu. V., “Lineinye formy i statisticheskie kriterii. I, II”, Ukr. matem. zh., 5:2 (1953), 207–243; 3, 247–290

[8] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1984, 752 pp.

[9] Bening V. E., Korolev V. Yu., Generalized Poisson Models and Their Applications in Insurance and Finance, VSP, Utrecht, 2002, 434 pp. | Zbl

[10] Bernstein S. N., “Sur les fonctions absolument monotones”, Acta Mathem., 52 (1928), 1–66 | DOI | MR | Zbl

[11] Bondesson L., “A general result on infinite divisibility”, Ann. Probab., 7:6 (1979), 965–979 | DOI | MR | Zbl

[12] Box G., Tiao G., Bayesian Inference in Statistical Analysis, Addison–Wesley, Reading, 1973, 588 pp. | MR | Zbl

[13] Bunge J., “Compositions semigroups and random stability”, Ann.Probab., 24 (1996), 1476–1489 | DOI | MR | Zbl

[14] Choy S. T. B., Smith A. F. M., “Hierarchical models with scale mixtures of normal distributions”, TEST, 6 (1997), 205–221 | DOI | Zbl

[15] Choy S. T. B., “Contribution to the discussion of Robustifying Bayesian procedures”, Bayesian Statistics, 6, eds. S. G. Walker, E. Guttierez-Pena, J. M. Bernardo, J. O. Berger, A. P. Dawid, A. F. M. Smith, eds., Oxford University Press, New York, 1999, 685–710 | MR

[16] Choy S. T. B., Chan C. M., “Scale mixtures distributions in insurance applications”, Astin Bull., 33:1 (2003), 93–104 | DOI | MR | Zbl

[17] Devroye L., “A note on Linnik's distribution”, Statist. Probab. Lett., 9:4 (1990), 305–306 | DOI | MR | Zbl

[18] Evans M., Hastings N., Peacock B., Statistical Distributions, 3, Wiley, New York, 2000 | MR | Zbl

[19] Gleser L. J., “The gamma distribution as a mixture of exponential distributions”, The American Statist., 43:2 (1989), 115–117 | MR

[20] Gnedenko B. V., Korolev V. Yu., Random Summation: Limit Theorems and Applications, CRC Press, Boca Raton, 1996, 288 pp. | MR | Zbl

[21] Gnedenko B. V., Kovalenko I. N., Introduction to Queueing Theory, Israel Program for Scientific Translations, Jerusalem, 1968, 281 pp. | MR

[22] Gnedenko B. V., Kovalenko I. N., Introduction to Queueing Theory, 2, Birkhauser, Boston, 1989, 315 pp. | MR

[23] Goldie C. M., “A class of infinitely divisible distributions”, Proc. Cambridge Philos. Soc., 63 (1967), 1141–1143 | DOI | MR | Zbl

[24] Gorenflo R., Kilbas A. A., Mainardi F., Rogosin S. V., Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014, 441 pp. | MR | Zbl

[25] Gorenflo R., Mainardi F., “Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects”, Chap. 4, Anomalous Transport: Foundations and Applications, eds. R. Klages, G. Radons, I. M. Sokolov, Wiley-VCH, Weinheim, Germany, 2008, 93–127 | DOI

[26] Grandell J., Mixed Poisson Processes, Chapman and Hall, London, 1997, 268 pp. | MR | Zbl

[27] Kalashnikov V. V., Geometric Sums: Bounds for Rare Events with Applications, Kluwer Academic Publishers, Dordrecht, 1997, 270 pp. | MR | Zbl

[28] Klebanov L. B., Rachev S. T., “Sums of a random number of random variables and their approximations with $\varepsilon$-accompanying infinitely divisible laws”, Serdica, 22 (1996), 471–498 | MR

[29] Korolev V. Yu., “Product representations for random variables with the Weibull distributions and their applications”, J. Math. Sci., 218:3 (2016), 298–313 | DOI | MR | Zbl

[30] Korolev V. Yu., Zeifman A. I., “A note on mixture representations for the Linnik and Mittag-Leffler distributions and their applications”, J. Mathem. Sci., (New York), 218:3 (2016), 314–327 | DOI | MR | Zbl

[31] Korolev V. Yu., Zeifman A. I., “Convergence of statistics constructed from samples with random sizes to the Linnik and Mittag-Leffler distributions and their generalizations”, J. Korean Statist. Soc. (to appear)

[32] Kotz S., Ostrovskii I. V., Hayfavi A., “Analytic and asymptotic properties of Linnik's probability densities. I”, J. Mathem. Anal. Appl., 193:1 (1995), 353–371 | DOI | MR | Zbl

[33] Kotz S., Ostrovskii I. V., Hayfavi A., “Analytic and asymptotic properties of Linnik's probability densities. II”, J. Mathem. Anal. Appl., 193:2 (1995), 497–521 | DOI | MR | Zbl

[34] Kotz S., Ostrovskii I. V., “A mixture representation of the Linnik distribution”, Statist. Probab. Lett., 26:1 (1996), 61–64 | DOI | MR | Zbl

[35] Laha R. G., “On a class of unimodal distributions”, Proc. Amer. Mathem. Soc., 12 (1961), 181–184 | DOI | MR | Zbl

[36] Pillai R. N., “Harmonic mixtures and geometric infinite divisibility”, J. Indian Statist. Assoc., 28 (1990), 87–98 | MR

[37] Pillai R. N., “On Mittag-Leffler functions and related distributions”, Ann. Inst. Statist. Math., 42:1 (1990), 157–161 | DOI | MR | Zbl

[38] Schneider W. R., “Stable distributions: Fox function representationand generalization”, Stochastic Processes in Classical and Quantum Systems, eds. S. Albeverio, G. Casati, D. Merlini, Springer, Berlin, 1986, 497–511 | DOI | MR

[39] Subbotin M. T., “On the law of frequency of error”, Matem. sb., 31:2 (1923), 296–301 | Zbl

[40] Tucker H., “On moments of distribution functions attracted to stable laws”, Houston J. Math., 1:1 (1975), 149–152 | MR | Zbl

[41] Uchaikin V. V., Zolotarev V. M., Chance and Stability, VSP, Utrecht, 1999, 570 pp. | MR | Zbl

[42] Walker S. G., Guttierez-Pena E., “Robustifying Bayesian procedures (with discussion)”, Bayesian Statistics, 6, eds. J. M. Bernardo, J. O. Berger, A. P. Dawid, A. F. M. Smith, Oxford University Press, New York, 1999, 685–710 | MR | Zbl

[43] Weron K., Kotulski M., “On the Cole-Cole relaxation function and related Mittag-Leffler distributions”, Physica A, 232 (1996), 180–188 | DOI