## Bernoulli

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

### Domains of attraction on countable alphabets

#### 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