Limit theorems and phase transitions for two models of summation of i.i.d. random variables with a parameter
Teoriâ veroâtnostej i ee primeneniâ, Tome 59 (2014) no. 2, pp. 340-364 Cet article a éte moissonné depuis la source Math-Net.Ru

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     title = {Limit theorems and phase transitions for two models of summation of i.i.d. random variables with a parameter},
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M. Grabchak; S. A. Molchanov. Limit theorems and phase transitions for two models of summation of i.i.d. random variables with a parameter. Teoriâ veroâtnostej i ee primeneniâ, Tome 59 (2014) no. 2, pp. 340-364. http://geodesic.mathdoc.fr/item/TVP_2014_59_2_a6/

[1] Abramovits M., Stigan I., Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979, 830 pp.

[2] Barczy M., Pap G., “Portmanteau theorem for unbounded measures”, Statist. Probab. Lett., 76:17 (2006), 1831–1835 | DOI | MR | Zbl

[3] Ben Arous G., Bogachev L. V., Molchanov S. A., “Limit theorems for sums of random exponentials”, Probab. Theory Related Fields, 132:4 (2005), 579–612 | DOI | MR | Zbl

[4] Bianchi M. L., Rachev S. T., Kim Y. S., Fabozzi F. J., “Tempered infinitely divisible distributions and processes”, Teoriya veroyatn. i ee primen., 55:1 (2011), 59–86 | DOI | MR | Zbl

[5] Billingsley P., Probability and Measure, Wiley, New York, 1995, 593 pp. | MR | Zbl

[6] Bingham N. H., Goldie C. M., Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge, 1987, 491 pp. | MR | Zbl

[7] Bogachev L. V., “Extreme value theory for random exponentials”, Probability and Mathematical Physics, A Volume in Honor of Stanislav Molchanov, CRM Proc. Lecture Notes, 42, eds. D. A. Dawson, V. Jaksic, B. Vainberg, Amer. Math. Soc., Providence, 2007, 41–64 | MR | Zbl

[8] Bovier A., Kurkova I., Löwe M., “Fluctuations of the free energy in the {REM} and the $ p$-spin {SK} model”, Ann. Probab., 30:2 (2002), 605–651 | DOI | MR | Zbl

[9] Brockmann D., Hufnagel L., “The scaling law of human travel — A message from George”, Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics, eds. B. Blasius, J. Kurths, L. Stone, World Scientific Publ., Singapore, 2007, 109–127 | DOI

[10] Bullock J. M., Clarke R. T., “Long distance seed dispersal by wind: measuring and modelling the tail of the curve”, Oecologia, 124:4 (2000), 506–521 | DOI

[11] Chakrabarty A., Meerschaert M. M., “Tempered stable laws as random walk limits”, Statist. Probab. Lett., 81:8 (2011), 989–997 | DOI | MR | Zbl

[12] Chakrabarty A., Samorodnitsky G., Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not?, Stoch. Models, 12:1 (2012), 109–143 | DOI | MR

[13] Cont R., Tankov P., Financial Modelling with Jump Processes, Chapman Hall, Boca Raton, 2004, 535 pp. | MR

[14] Cranston M., Molchanov S. A., “Limit laws for sums of products of exponentials of i.i.d. random variables”, Israel J. Math., 148:1 (2005), 115–136 | DOI | MR | Zbl

[15] Derrida B., “Random-energy model: Limit of a family of disordered models”, Phys. Rev. Lett., 45:2 (1980), 79–82 | DOI | MR

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

[17] Feng Y., Molchanov S., Whitmeyer J., “Random walks with heavy tails and limit theorems for branching processes with migration and immigration”, Stoch. Dyn., 12:1 (2012), 1150007, 23 pp. | DOI | MR | Zbl

[18] Grabchak M., Limit theorems for sequences of tempered stable and related distributions, arXiv: 1201.6006

[19] Grabchak M., “On a new class of tempered stable distributions: moments and regular variation”, J. Appl. Probab., 49:4 (2012), 1015–1035 | DOI | MR | Zbl

[20] Grabchak M., “Inversions of Lévy measures and the relation between long and short time behavior of Lévy processes”, J. Theoret. Probab., 2013

[21] Grabchak M., Malchanov S. A., “Predelnye teoremy i fazovye perekhody dlya dvukh modelei summirovaniya nezavisimy odinakovo raspredelennykh sluchainykh velichin, zavisyaschikh ot parametrov”, Dokl. RAN, 451:4 (2013), 374–377 | DOI | Zbl

[22] Grabchak M., Samorodnitsky G., “Do financial returns have finite or infinite variance? A paradox and an explanation”, Quant. Finance, 10:8 (2010), 883–893 | DOI | MR | Zbl

[23] Gut A., Stopped Random Walks: Limit Theorems and Applications, Springer, New York, 2009, 263 pp. | MR

[24] Ibragimov I. A., Linnik Yu. V., Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Gröningen, 1971, 443 pp. | MR | Zbl

[25] Ibragimov I. A., Presman E. L., “O skorosti sblizheniya raspredelenii summ nezavisimykh sluchainykh velichin s soprovozhdayuschimi zakonami”, Teoriya veroyatn. i ee primen., 18:4 (1973), 753–766 | MR

[26] Jørgensen B., Martinez J. R., Vinogradov V., “Domains of attraction to Tweedie distributions”, Lith. Math. J., 49:4 (2009), 399–425 | DOI | MR

[27] Kondratiev Y., Kutoviy O., Pirogov S., “Correlation functions and invariant measures in continuous contact model”, Infin. Dimens. Anal., Quantum Probab. Relat. Top., 11:2 (2008), 231–258 | DOI | MR | Zbl

[28] Le Cam L., “On the distribution of sums of independent random variables”, Bernoulli 1713 Bayes 1763 Laplace 1813: Anniversary Volume, eds. J. Neyman, L. Le Cam, Berlin–New York, 1965, 179–202 | MR | Zbl

[29] Meerschaert M. M., Scheffler H., Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, Wiley, Chichester, 2001, 484 pp. | MR | Zbl

[30] Resnick S. I., Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Springer, New York, 2007, 404 pp. | MR | Zbl

[31] Rosiński J., “Tempering stable processes”, Stochastic Process. Appl., 117:6 (2007), 677–707 | DOI | MR | Zbl

[32] Sato K., Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge, 1999, 486 pp. | MR | Zbl

[33] Zaitsev A. Yu., Arak T. V., “O skorosti skhodimosti vo vtoroi ravnomernoi predelnoi teoreme Kolmogorova”, Teoriya veroyatn. i ee primen., 28:2 (1983), 333–353 | MR | Zbl

[34] Zaitsev A. Yu., “O ravnomernoi approksimatsii funktsii raspredeleniya summ nezavisimykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 32:1 (1987), 45–52 | MR

[35] Zaitsev A. Yu., “Mnogomernyi variant vtoroi ravnomernoi predelnoi teoremy Kolmogorova”, Teoriya veroyatn. i ee primen., 34:1 (1989), 128–151 | MR

[36] Zaitsev A. Yu., “Approksimatsiya svertok veroyatnostnykh raspredeleniya bezgranichno delimymi zakonami pri oslablennykh momentnykh ogranicheniyakh”, Zap. nauchn. sem. LOMI, 194, 1992, 79–90