Geodesic
Probability generating functions for discrete real-valued random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 129-149 Cet article a éte moissonné depuis la source Math-Net.Ru

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The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real-valued discrete random variables, the well-known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5, allow us to extend this technique to general discrete random variables. Being so, the purpose of this work is twofold. First, we show that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions. Second, we intend to draw attention to some practical ways of performing the necessary calculations.
Keywords: probability generating functions, finite sums of independent real-valued discrete random variables, Dirichlet series. 
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M. L. Esquível. Probability generating functions for discrete real-valued random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 129-149. http://geodesic.mathdoc.fr/item/TVP_2007_52_1_a7/

[1] Dowling M. M., Nakamura M., “Estimating parameters for discrete distributions via the empirical probability generating function”, Comm. Statist. Simulation Comput., 26:1 (1997), 301–313 | DOI | Zbl

[2] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 1, Mir, M., 1984, 527 pp.; 752 pp.

[3] Hardy G. H., Riesz M., The General Theory of Dirichlet's Series, Cambridge Univ. Press, 1915 | Zbl

[4] Hoffmann-Jørgensen J., Probability with a View towards Statistics, V. I, Chapman Hall, New York, 1994, 589 pp.

[5] Kahane J.-P., “The last problem of Harald Bohr”, J. Aust. Math. Soc., A47 (1989), 133–152 | DOI | MR

[6] Kallenberg O., Foundations of Modern Probability, Springer-Verlag, New York, 2002, 638 pp. | MR

[7] Lando S. K., Lektsii o proizvodyaschikh funktsiyakh, MTsNMO, M., 2002, 143 pp.

[8] Marques M. S., Pérez-Abreu V., “Law of large numbers and central limit theorem for the empirical probability generating function of stationary random sequences and processes”, Symposium on Probability Theory and Stochastic Processes (Guanajuato, 1988), Aportaciones Mat. Notas Investigación, 4, eds. M. E. Caballero et al., Soc. Mat. Mexicana, México, 1989, 100–109 | MR | Zbl

[9] Nakamura M., Pérez-Abreu V., “Empirical probability generating function: An overview”, Insurance Math. Econom., 12:3 (1993), 287–295 | DOI | Zbl

[10] Nakamura M., Pérez-Abreu V., “Exploratory data analysis for counts using the empirical probability generating function”, Comm. Statist. Theory Methods, 22:3 (1993), 827–842 | DOI | Zbl

[11] Nakamura M., Pérez-Abreu V., “Use of an empirical probability generating function for testing a Poisson model”, Canad. J. Statist., 21:2 (1993), 149–156 | DOI | MR | Zbl

[12] Rémillard B., Theodorescu R., “Inference based on the empirical probability generating function for mixtures of Poisson distributions”, Statist. Decisions, 18:4 (2000), 349–366 | MR | Zbl

[13] Resnick S. I., A Probability Path., Birkhäuser, Boston, 1999, 453 pp. | MR | Zbl

[14] Rudin W., Real and Complex Analysis, McGraw-Hill, New York, 1987, 416 pp. | MR | Zbl

[15] Rueda R., Pérez-Abreu V., O'Reilly F., “Goodness of fit for the Poisson distribution based on the probability generating function”, Comm. Statist. Theory Methods, 20:10 (1991), 3093–3110 | DOI | MR | Zbl

[16] Rueda R., O'Reilly F., “Tests of fit for discrete distributions based on the probability generating function”, Comm. Statist. Simulation Comput., 28:1 (1999), 259–274 | DOI | MR | Zbl

[17] Shiryaev A. N., Veroyatnost, t. 1, 2, MTsNMO, M., 2004, 927 pp.

[18] Saks S., Zygmund A., Analytic Functions, Monografie Matematyczne, 28, Polskie Towarzystvo Matematyczne, Warszawa–Wroclaw, 1952, 451 pp. | MR