Domains of attraction on countable alphabets (Q2405205)

Consider an alphabet with conutably many letters \(\mathcal{X} = \{l_k, k \geq 1\}\) and an associated probability distribution \(P = \{p_k, k \geq 1\} \in \mathcal{P}\), where \(\mathcal{P}\) is the class of all probability distributions on \(\mathcal {X}\). Defining \(\tau_n = \sum_{k \geq 1} n p_k (1 - p_k)^n\), the author classifies \(P\in \mathcal{X}\) to belong to (i) Domain 0 (no tails) if \(\lim_{n \rightarrow \infty} \tau_n = 0\), (ii) Domain 1 (thin tails) if \(\limsup_{n \rightarrow \infty} \tau_n \) equal to a positive constant, (iii) Domain 2 (thick tails) if \(\lim_{n \rightarrow \infty} \tau_n = \infty\), and (iv) Domain \(T\) if it does not belong to Domains 0, 1 or 2, and the four domains form a partition of \(\mathcal{P}\). The author shows that Domain 0 does and only does include probability distributions with positive probabilities on a finite subset of \(\mathcal{X}\); Domain 1 includes distributions with thin tails such as \(p_k \alpha a^{- \lambda k}, \) \(p_k \alpha a^{- \lambda k^2}, \) and \(p_k \alpha k^r a^{- \lambda k}, \) where \( a > 1, \lambda > 0, r \) a real number; Domain 2 includes distributions such as \(p_k \alpha k^{- \lambda } \) and \(p_k \alpha (k \ln^{\lambda} k)^{-1}, \lambda > 1\); Domain \(T\) is non-empty; and that under a regularity condition, all distributions on a countably infinite alphabet that are dominated by a Domain 1 distribution must also belong to Domain 1. The author also establishes that in Domain 0, \(\tau_n \rightarrow 0\) exponentially fast for every distribution; the tail index \(\tau_n\) of a distribution with tail \(p_k \alpha e^{- \lambda k}, \lambda > 0\), in Domain 1 oscillates between two positive constants and does not have a limit as \(n \rightarrow \infty\), and there is a uniform positive lower bound for \(\limsup_{n \rightarrow \infty} \tau_n\) for all distributions with positive probabilities on infinitely many letters of \(\mathcal{X}\). Several constructed examples are given to illustrate the results. There is a brief discussion on the statistical implication of the established results, along with several remarks. To honour earlier researchers whose work led to the results of this article, the author wishes to have the three domains 0, 1 and 2, respectively, to be identified as the Gini-Simpson family, the Molchanov family and the Turing-Good family.