Geodesic
Limit theorems for some classes of power series type distributions
Diskretnaya Matematika, Tome 30 (2018) no. 4, pp. 134-145.

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Several classes of distributions of power series type with finite and infinite radii of convergence are considered. For such distributions local limit theorems are obtained as the parameter of distribution tends to the right end of the interval of convergence. For the case when the convergence radius equals to 1, we prove an integral limit theorem on the convergence of distributions of random variables $(1-x)\xi_x$as $x\to 1-$ to the gamma-distribution ($\xi_x$ is a random variable with corresponding distribution of the power series type). The proofs are based on the steepest descent method.
Keywords: power series distribution, local limit theorems, radius of convergence, steepest descent method. 
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     author = {A. N. Timashev},
     title = {Limit theorems for some classes of power series type distributions},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2018_30_4_a11/}
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A. N. Timashev. Limit theorems for some classes of power series type distributions. Diskretnaya Matematika, Tome 30 (2018) no. 4, pp. 134-145. http://geodesic.mathdoc.fr/item/DM_2018_30_4_a11/

[1] Noack A., “A class of random variables with discrete distributions”, Ann. Math. Statist., 21:1 (1950), 127–132 | DOI | MR | Zbl

[2] Dzhonson N. L., Kots S., Kemp A., Odnomernye diskretnye raspredeleniya, Binom. Laboratoriya znanii, M., 2010, 560 pp.; Johnson N.L., Kotz S., Kemp A.W., Univariate discrete distributions, Second Edition, John Wiley Sons, 1992, 565 pp. | MR | Zbl

[3] Timashev A. N., Raspredeleniya tipa stepennogo ryada i obobschennaya skhema razmescheniya, ID «Akademiya», M., 2016, 168 pp.

[4] Timashev A. N., “Predelnye teoremy dlya raspredelenii tipa stepennogo ryada s konechnym radiusom skhodimosti”, Teoriya veroyatn. i ee primen., 63:1 (2018), 57–69 | DOI | MR | Zbl

[5] Timashev A. N., Additivnye zadachi s ogranicheniyami na znacheniya slagaemykh, ID «Akademiya», M., 2015, 184 pp.

[6] Evgrafov M. A., Asimptoticheskie otsenki i tselye funktsii, Fizmatgiz, M., 1962, 320 pp. | MR

[7] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967; Bateman H., Erdélyi A., Higher transcendental functions, McGraw-Hill, 1953

[8] Timashev A. N., Asimptoticheskie razlozheniya v veroyatnostnoi kombinatorike, TVP, M., 2011, 312 pp.

[9] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1967 ; Feller W., An Introduction to Probability Theory and Its Applications, v. 2, John Wiley Sons, 1966, 626 pp. | MR | MR | Zbl

[10] Chandrasekkharan K., Arifmeticheskie funktsii, Nauka, M., 1975, 272 pp. ; Chandrasekharan K., Arithmetical functions, Springer-Verlag, 1970, 245 pp. | MR | MR | Zbl

[11] Kolchin V. F., Sluchainye grafy, Fizmatlit, M., 2000, 256 pp.; Kolchin V. F., Random Graphs, Cambridge Univ. Press, 1998, 268 pp. | MR