Limit laws for exponential families
type: Item
instance of: scholarly article
title: Limit laws for exponential families
zbMATH Open document ID: 0939.62020
DOI: 10.2307/3318554
author: August A. Balkema
author: Claudia Klüppelberg
author: Sidney I. Resnick
published in: Bernoulli
publication date: 2000-07-03T00:00:00Z
full work available at URL: document.pdf
review text: The distribution function \(F\) of the random variable \(X\) generates a natural exponential family of distribution functions \(\{F_\lambda, \lambda\in \Lambda\}\), where \[ dF_\lambda(x): =e^{\lambda x}dF(x)/Ee^{\lambda x}\quad\text{for}\quad \lambda\in\Lambda: =\bigl\{\lambda\in(-\infty,\infty) \mid Ee^{\lambda X}<\infty\bigr\}. \] The authors study the asymptotic behaviour of the distribution function \(F_\lambda\) as \(\lambda\) increases to \(\lambda_\infty: =\sup \Lambda\). One of the main results states as follows: Assume \(\lambda_\infty \leq\infty\) does not lie in \(\Lambda\). Then the only possible non-degenerate limit laws for the normalised distributions \(F_\lambda (a_\lambda x+b_\lambda)\) are the normal distribution and the gamma distributions on \((0,\infty)\) and on \((-\infty,0)\).
reviewed by: Wiesław Dziubdziela
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1395957
zbMATH Keywords: asymptotic normality
zbMATH Keywords: limit laws
zbMATH Keywords: gamma distributions
MaRDI profile type: Publication
OpenAlex ID: W1998343004
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description: scientific article; zbMATH DE number 1395957