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Nonconventional limits of random sequences related to partitions of integers


Authors: Jordan M. Stoyanov and Christophe Vignat
Journal: Proc. Amer. Math. Soc. 148 (2020), 1791-1804
MSC (2010): Primary 60E10, 11B68, 60F05, 60C99
DOI: https://doi.org/10.1090/proc/14638
Published electronically: January 6, 2020
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Abstract | References | Similar Articles | Additional Information

Abstract: We deal with a sequence of integer-valued random variables
$ \{Z_N\}_{N=1}^{\infty }$ which is related to restricted partitions, or representations, of
positive integers. We observe that $ Z_N=X_1+ \cdots + X_N$ for independent and bounded random variables $ X_j$'s, so $ Z_N$ has finite mean $ \mathbf {E}Z_N$ and
variance $ \mathbf {Var}Z_N$. We want to find the limit distribution of $ {\hat Z}_N=\linebreak\left (Z_N-\mathbf {E}Z_N\right )/{\sqrt {\mathbf {Var}Z_N}}$ as $ N \to \infty .$ While in many cases the limit distribution is normal, the main results established in this paper are that $ {\hat Z}_N \overset {d}{\to } Z_{*},$ where $ Z_{*}$ is a bounded random variable. We find explicitly the range of values of $ Z_*$ and derive some properties of its distribution. The main tools used are moment generating functions, cumulant generating functions, moments, and cumulants of the random variables involved. Useful related topics are also discussed.


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Additional Information

Jordan M. Stoyanov
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
Email: stoyanovj@gmail.com

Christophe Vignat
Affiliation: L.S.S., CentraleSupelec, Université Paris-Sud XI, Orsay, France; and Department of Mathematics, Tulane University, New Orleans, Lousiana 70118
Email: cvignat@tulane.edu

DOI: https://doi.org/10.1090/proc/14638
Received by editor(s): March 4, 2019
Published electronically: January 6, 2020
Additional Notes: The first named author acknowledges the support provided by Academia Sinica, Institute of Statistical Science (Taiwan, RoC), for a research visit, November 2018, when the present paper was at the last stage of completion and the results were presented publicly at a seminar talk.
The material in this paper is partly based upon work by the second named author, who was supported by the NSF (Grant No. DMS-1439786) to visit the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Point Configurations in Geometry, Physics and Computer Science semester program, Spring 2018.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society