Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T09:47:11.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Normality Through Factorial Cumulants and Partition Identities

Published online by Cambridge University Press:  21 December 2012

KONSTANCJA BOBECKA ,
PAWEŁ HITCZENKO ,
FERNANDO LÓPEZ-BLÁZQUEZ ,
GRZEGORZ REMPAŁA  and
JACEK WESOŁOWSKI
KONSTANCJA BOBECKA
Affiliation:
Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Warszawa, Poland (e-mail: bobecka@mini.pw.edu.pl, wesolo@mini.pw.edu.pl)
PAWEŁ HITCZENKO
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, USA (e-mail: phitczenko@math.drexel.edu)
FERNANDO LÓPEZ-BLÁZQUEZ
Affiliation:
Facultad de Matemáticas Universidad de Sevilla, Sevilla, Spain (e-mail: lopez@us.es)
GRZEGORZ REMPAŁA
Affiliation:
Department of Biostatistics, Georgia Health University, Augusta, USA (e-mail: grempala@georgiahealth.edu)
JACEK WESOŁOWSKI
Affiliation:
Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Warszawa, Poland (e-mail: bobecka@mini.pw.edu.pl, wesolo@mini.pw.edu.pl)
Get access Rights & Permissions [Opens in a new window]

Abstract

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for ‘moments’ of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.

Keywords

Primary 60F05Secondary 60E10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 213 - 240
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bates, G. E. and Neyman, J. (1952) Contributions to the theory of accident proneness I: An optimistic model of the correlation between light and severe accidents. Univ. California Publ. Statist. 1 215–253.Google Scholar
[2]Billingsley, P. (1995) Probability and Measure, third edition, Wiley.Google Scholar
[3]Chuprunov, A. and Fazekas, I. (2010) An inequality for moments and its applications to the 541 generalized allocation scheme. Publ. Math. Debrecen 76 271–286.CrossRefGoogle Scholar
[4]Goodhart, G. J., Ehrenberg, A. S. C. and Chatfield, C. (1984) The Dirichlet: A comprehensive model of buying behaviour. J. Royal Statist. Soc. Section A 147 621–655.CrossRefGoogle Scholar
[5]Graham, R. L., Knuth, D. E. and Patashnik, O. (1994) Concrete Mathematics, Addison-Wesley.Google Scholar
[6]Haldane, J. B. S. (1945) On a method of estimating frequencies. Biometrika 33 222–225.CrossRefGoogle ScholarPubMed
[7]Hwang, H-K. and Janson, S. (2008) Local limit theorems for finite and infinite urn models. Ann. Probab. 36 992–1022.Google Scholar
[8]Janson, S.Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience.Google Scholar
[9]Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997) Discrete Multivariate Distributions, Wiley Series in Probability and Statistics, Wiley.Google Scholar
[10]Johnson, N. L., Kotz, S. and Kemp, A. W. (1992) Univariate Discrete Distributions, second edition, Wiley Series in Probability and Statistics, Wiley.Google Scholar
[11]Keener, R. W. and Wu, W. B. (2006) On Dirichlet multinomial distributions. In Random Walk, Sequential Analysis and Related Topics, World Scientific, pp. 118–130.Google Scholar
[12]Kendall, M. G. and Stuart, A. (1969) The Advanced Theory of Statistics, Vol. 1: Distribution Theory. Addison-Wesley.Google Scholar
[13]Kolchin, V. F. (1968) A certain class of limit theorems for conditional distributions. Litovsk. Mat. Sb. 8 53–63.Google Scholar
[14]Kolchin, V. F. (1999) Random Graphs. Vol. 53 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.Google Scholar
[15]Kolchin, V. F. (1986) Random Mappings, Optimization Software.Google Scholar
[16]Kolchin, V. F., Sevastyanov, B. A. and Chistyakov, V. P. (1978) Random Allocations, Winston.Google Scholar
[17]Letac, G. and Mora, M. (1990) Natural real exponential families with cubic variance functions. Ann. Statist. 18 1–37.Google Scholar
[18]Pavlov, Y. L. (1977) Limit theorems for the number of trees of a given size in a random forest. Mat. Sbornik 103 335–345.Google Scholar
[19]Pavlov, Y. L. (2000) Random Forests, VSP.CrossRefGoogle Scholar
[20]Stanley, R. P. (1999) Enumerative Combinatorics, Vol. 2, Cambridge University Press.CrossRefGoogle Scholar
[21]Stanley, R. P. (2009) Personal communication.Google Scholar
[22]van der Vaart, A. W. (1998) Asymptotic Statistics, Cambridge University Press.CrossRefGoogle Scholar
[23]Whittaker, E. T. and Watson, G. N. (1996) A Course of Modern Analysis, fourth edition, Cambridge University Press.Google Scholar