Bernoulli

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

Limit laws for exponential families

August A. Balkema, Claudia Klüppelberg, and Sidney I. Resnick

Full-text: Open access

Abstract

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

Λ :={λ:K(λ):=rmErme λ X<}.

The distribution F generates a natural exponential family of distribution functions { F λ ,λΛ} , where

rm dF λ (x):=rme λ xrmdF(x)/K(λ),λΛ.

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1143122297

Mathematical Reviews number (MathSciNet)
MR1735779

Zentralblatt MATH identifier
0939.62020

Keywords
affine transformation asymptotic normality convergence of types cumulant generating function exponential family Esscher transform gamma distribution Gaussian tail limit law normal distribution moment generating function power norming semistable stochastically compact universal distributions

Citation

Balkema, August A.; Klüppelberg, Claudia; Resnick, Sidney I. Limit laws for exponential families. Bernoulli 5 (1999), no. 6, 951--968. https://projecteuclid.org/euclid.bj/1143122297


Export citation