Open Access
May 2018 Domains of attraction on countable alphabets
Zhiyi Zhang
Bernoulli 24(2): 873-894 (May 2018). DOI: 10.3150/15-BEJ786


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.


Download Citation

Zhiyi Zhang. "Domains of attraction on countable alphabets." Bernoulli 24 (2) 873 - 894, May 2018.


Received: 1 April 2015; Revised: 1 August 2015; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778350
MathSciNet: MR3706779
Digital Object Identifier: 10.3150/15-BEJ786

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

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 2 • May 2018
Back to Top