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

Limit laws for exponential families

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

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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

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

First available in Project Euclid: 23 March 2006

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Zentralblatt MATH identifier

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


Balkema, August A.; Klüppelberg, Claudia; Resnick, Sidney I. Limit laws for exponential families. Bernoulli 5 (1999), no. 6, 951--968.

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