## Bernoulli

• Bernoulli
• Volume 5, Number 6 (1999), 951-968.

### Limit laws for exponential families

#### Abstract

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

$Λ :={λ∈ℝ:K(λ):=rmErme λ X<∞}.$

The distribution $F$ generates a natural exponential family of distribution functions ${ F λ ,λ∈Λ}$, where

$rm dF λ (x):=rme λ xrmdF(x)/K(λ),λ∈Λ.$

We study the asymptotic behaviour of the distribution functions $F λ$ as $λ$ increases to $λ ∞ :=supΛ$. If $λ ∞ =∞$ then $F λ ↓0$ pointwise on ${ F<1}$. It may still be possible to obtain a non-degenerate weak limit law $G (y)=limF λ(a λy+b λ)$ by choosing suitable scaling and centring constants $a λ >0$ and $b λ$, and in this case either $G$ is a Gaussian distribution or $G$ has a finite lower end-point $y 0 =inf{G>0}$ and $G (y-y 0)$ is a gamma distribution. Similarly, if $λ ∞$ is finite and does not belong to $Λ$ then $G$ is a Gaussian distribution or $G$ has a finite upper end-point $y ∞$ and $1 -G(y ∞-y)$ is a gamma distribution. The situation for sequences $λ n ↑λ ∞$ is entirely different: any distribution function may occur as the weak limit of a sequence $F λ n (a nx+b n)$.

#### Article information

Source
Bernoulli, Volume 5, Number 6 (1999), 951-968.

Dates
First available in Project Euclid: 23 March 2006