• Bernoulli
  • Volume 24, Number 2 (2018), 873-894.

Domains of attraction on countable alphabets

Zhiyi Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For each probability distribution on a countable alphabet, a sequence of positive functionals are developed as tail indices. By and only by the asymptotic behavior of these indices, domains of attraction for all probability distributions on the alphabet are defined. The three main domains of attraction are shown to contain distributions with thick tails, thin tails and no tails respectively, resembling in parallel the three main domains of attraction, Fréchet, Gumbel and Weibull families, for continuous random variables on the real line. In addition to the probabilistic merits associated with the domains, the tail indices are partially motivated by the fact that there exists an unbiased estimator for every index in the sequence, which is therefore statistically observable, provided that the sample is sufficiently large.

Article information

Bernoulli, Volume 24, Number 2 (2018), 873-894.

Received: April 2015
Revised: August 2015
First available in Project Euclid: 21 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

distributions on alphabets domains of attraction tail index Turing’s formula


Zhang, Zhiyi. Domains of attraction on countable alphabets. Bernoulli 24 (2018), no. 2, 873--894. doi:10.3150/15-BEJ786.

Export citation


  • [1] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [2] Fisher, R.A. and Tippett, L.H.C. (1928). Limiting forms of the frequency-distribution of the largest or smallest member of a sample. Math. Proc. Cambridge Philos. Soc. 24 180.
  • [3] Fréchet, M. (1927). Sur la loi de probabilité de l’écart maximum. Annals de la Soc. Polonaise. de Math. 6 92.
  • [4] Gini, C. (1912). Variabilità e Mutabilità. Contributo Allo Studiodelle Distribuzioni e delle Relazioni Statistiche. Bologna: C. Cuppini.
  • [5] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. (2) 44 423–453.
  • [6] Good, I.J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika 40 237–264.
  • [7] Simpsom, E.H. (1949). Measurement of diversity. Nature 163 688.
  • [8] Smirnov, N.V. (1952). Limit Distributions for the Terms of a Variational Series. Am. Math. Soc., Providence, RI.
  • [9] Zhang, Z. (2012). Entropy estimation in Turing’s perspective. Neural Comput. 24 1368–1389.
  • [10] Zhang, Z. and Grabchak, M. (2016). Entropic representation and estimation of Diversity indices. J. Nonparametr. Stat. 28 563–575.
  • [11] Zhang, Z. and Zhou, J. (2010). Re-parameterization of multinomial distributions and diversity indices. J. Statist. Plann. Inference 140 1731–1738.