Domains of Attraction on Countable Alphabets

TL;DR

This paper introduces tail index-based domains of attraction for probability distributions on countable alphabets, paralleling classical continuous domains, with statistically observable indices and implications for tail behavior classification.

Contribution

It develops a novel framework of tail indices for countable alphabets, defining domains of attraction analogous to classical continuous cases, with estimators for these indices.

Findings

Identifies three main domains of attraction: thick tails, thin tails, no tails.

Establishes the existence of unbiased estimators for tail indices.

Parallels classical Gumbel, Fréchet, Weibull domains for discrete distributions.

Abstract

For each probability distribution on a countable alphabet, a sequence of positive functionals are developed as tail indices based on Turing's perspective. 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, Gumbel, Frechet 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.