Limit laws for exponential families
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 \((-\infty,0)\).
scientific article; zbMATH DE number 1395957