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Bernard Candelpergher; Michel Miniconi
Une étude asymptotique probabiliste des coefficients d’une série entière
(A probabilistic asymptotic study of the coefficients of a power series)
Journal de théorie des nombres de Bordeaux, 26 no. 1 (2014), p. 45-67, doi: 10.5802/jtnb.858
Article PDF | Reviews MR 3232766 | Zbl 1296.60054

Résumé - Abstract

Following the ideas of Rosenbloom [7] and Hayman [5], Luis Báez-Duarte gives in [1] a probabilistic proof of Hardy-Ramanujan’s asymptotic formula for the partitions of an integer. The main principle of the method relies on the convergence in law of a family of random variables to a gaussian variable. In our work we prove a theorem of the Liapounov type (Chung [2]) that justifies this convergence. To obtain simple asymptotic formulæ a condition of the so-called strong Gaussian type defined by Luis Báez-Duarte is required; we demonstrate this in a situation that make it possible to obtain a classical asymptotic formula for the partitions of an integer with distinct parts (Erdös-Lehner [4], Ingham [6]).


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[2] K. L. Chung, A Course in Probability Theory, 3rd ed.. Academic Press, 2001.  MR 1796326 |  Zbl 0345.60003
[3] H. Cramér, Random Variables and Probability Distributions, 2nd ed.. Cambridge University Press, 1963.  MR 165599 |  Zbl 0016.36304 |  JFM 63.1080.01
[4] P. Erdös, J. Lehner, The Distribution of the Number of Summands in the Partitions of a Positive Integer. Duke Math. J. 8, 2 (1982), 335–345.  MR 4841 |  JFM 67.0126.02
[5] W. K. Hayman, A Generalisation of Stirling’s Formula. J. Reine Angew. Math. 196, 1/2 (1956), 67–95.  Zbl 0072.06901
[6] A. E. Ingham, A Tauberian Theorem for Partitions. Ann. of Math., 2nd series 42, 5 (1941), 1075–1090.  MR 5522 |  Zbl 0063.02973
[7] P. C. Rosenbloom, Probability and Entire Functions. Studies in Math. Analysis and Related Topics, Essays in Honor of G. Pólya, 45 (1962), 325–332.  MR 145074 |  Zbl 0112.30102