Limit laws for exponential families

Abstract

For a real random variable $X$ with distribution function $F$, define

$$\Lambda :=\{\lambda \in ℝ:K(\lambda ):=rmErm{e}^{\lambda X}<\mathrm{\infty }\}.$$

The distribution $F$ generates a natural exponential family of distribution functions $\{F_{\lambda },\lambda \in \Lambda \}$, where

$$rm{\mathrm{dF}}_{\lambda }\left(x\right):=rm{e}^{\lambda x}rm\mathrm{dF}\left(x\right)/K\left(\lambda \right),\lambda \in \Lambda .$$

We study the asymptotic behaviour of the distribution functions $F_{\lambda }$ as $\lambda$ increases to ${\lambda }_{\mathrm{\infty }}:=sup\Lambda$. If ${\lambda }_{\mathrm{\infty }}=\mathrm{\infty }$ then $F_{\lambda }\downarrow 0$ pointwise on $\left\{F<1\right\}$. It may still be possible to obtain a non-degenerate weak limit law $G\left(y\right)=lim{F}_{\lambda }(a_{\lambda }y+b_{\lambda })$ by choosing suitable scaling and centring constants $a_{\lambda }>0$ and $b_{\lambda }$, and in this case either $G$ is a Gaussian distribution or $G$ has a finite lower end-point $y_0=inf\left\{G>0\right\}$ and $G(y-y_0)$ is a gamma distribution. Similarly, if ${\lambda }_{\mathrm{\infty }}$ is finite and does not belong to $\Lambda$ then $G$ is a Gaussian distribution or $G$ has a finite upper end-point $y_{\mathrm{\infty }}$ and $1-G(y_{\mathrm{\infty }}-y)$ is a gamma distribution. The situation for sequences ${\lambda }_n\uparrow {\lambda }_{\mathrm{\infty }}$ is entirely different: any distribution function may occur as the weak limit of a sequence $F_{{\lambda }_n}(a_{\mathrm{nx}}+b_n)$.