Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: January 6, 2020
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [1] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 8th ed. Academic Press, 2015.
  • [2] Hsien-Kuei Hwang and Vytas Zacharovas, Limit distribution of the coefficients of polynomials with only unit roots, Random Structures Algorithms 46 (2015), no. 4, 707–738. MR 3346464,
  • [3] Svante Janson, Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs, Ann. Probab. 16 (1988), no. 1, 305–312. MR 920273
  • [4] Michel Loève, Probability theory. I, 4th ed., Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, Vol. 45. MR 0651017
  • [5] Kent E. Morrison, Random walks with decreasing steps, preprint, California Polytech. State Univ., 1998.
  • [6] OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences,, 2019.
  • [7] Valentin V. Petrov, Limit theorems of probability theory, Oxford Studies in Probability, vol. 4, The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables; Oxford Science Publications. MR 1353441
  • [8] Albert N. Shiryaev, Probability. 1, 3rd ed., Graduate Texts in Mathematics, vol. 95, Springer, New York, 2016. Translated from the fourth (2007) Russian edition by R. P. Boas and D. M. Chibisov. MR 3467826
  • [9] Jordan M. Stoyanov, Counterexamples in probability, Dover Publications, Inc., Mineola, NY, 2013. Third edition of [ MR0930671]; Revised, corrected and amended reprint of the second edition [ MR3444842]. MR 3837562
  • [10] Zhonggen Su, Random matrices and random partitions, World Scientific Series on Probability Theory and Its Applications, vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. Normal convergence. MR 3381296
  • [11] C. Vignat and T. Wakhare, Finite generating functions for the sum of digits sequence, Ramanujan J. (2018),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60E10, 11B68, 60F05, 60C99

Retrieve articles in all journals with MSC (2010): 60E10, 11B68, 60F05, 60C99

Additional Information

Jordan M. Stoyanov
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

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

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