Limit laws for exponential families

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

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

For a real random variable X with distribution function F, define A = {A ∈ ℝ : K(λ) := EeλX < ∞}. The distribution F generates a natural exponential family of distribution functions {Fλ, A ∈ A}, where dFλ(x) := eλXdF(x)/K(λ), λ ∈ A. We study the asymptotic behaviour of the distribution functions FI as A increases to A := sup A. If ASO = oo then FI 0 pointwise on {F< 1}. It may still be possible to obtain a non-degenerate weak limit law G(y) = lim Fjay + b) by choosing suitable scaling and centring constants a > 0 and b, and in this case either G is a Gaussian distribution or G has a finite lower end-point yo = inf{G>0} and G(y -y0) is a gamma distribution. Similarly, if A is finite and does not belong to A then G is a Gaussian distribution or G has a finite upper end-point yx and 1 -G(yx -y) is a gamma distribution. The situation for sequences A A is entirely different: any distribution function may occur as the weak limit of a sequence Fnm(ax+ b.

Original languageEnglish
Pages (from-to)951-968
Number of pages18
JournalBernoulli
Volume5
Issue number6
DOIs
StatePublished - 1999

Keywords

  • Affine transformation
  • Asymptotic normality
  • Convergence of types
  • Cumulant generating function
  • Esscher transform
  • Exponential family
  • Gamma distribution
  • Gaussian tail
  • Limit law
  • Normal distribution

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