Bernoulli

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

Domains of attraction on countable alphabets

Zhiyi Zhang

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1505980881

Digital Object Identifier
doi:10.3150/15-BEJ786

Mathematical Reviews number (MathSciNet)
MR3706779

Zentralblatt MATH identifier
06778350

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

Citation

Zhang, Zhiyi. Domains of attraction on countable alphabets. Bernoulli 24 (2018), no. 2, 873--894. doi:10.3150/15-BEJ786. https://projecteuclid.org/euclid.bj/1505980881


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