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 language | English |
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Pages (from-to) | 951-968 |
Number of pages | 18 |
Journal | Bernoulli |
Volume | 5 |
Issue number | 6 |
DOIs | |
State | Published - 1999 |
Keywords
- Affine transformation
- Asymptotic normality
- Convergence of types
- Cumulant generating function
- Esscher transform
- Exponential family
- Gamma distribution
- Gaussian tail
- Limit law
- Normal distribution