- Volume 5, Number 6 (1999), 951-968.
Limit laws for exponential families
For a real random variable with distribution function , define
The distribution generates a natural exponential family of distribution functions , where
We study the asymptotic behaviour of the distribution functions as increases to . If then pointwise on . It may still be possible to obtain a non-degenerate weak limit law by choosing suitable scaling and centring constants and , and in this case either is a Gaussian distribution or has a finite lower end-point and is a gamma distribution. Similarly, if is finite and does not belong to then is a Gaussian distribution or has a finite upper end-point and is a gamma distribution. The situation for sequences is entirely different: any distribution function may occur as the weak limit of a sequence .
Bernoulli, Volume 5, Number 6 (1999), 951-968.
First available in Project Euclid: 23 March 2006
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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. https://projecteuclid.org/euclid.bj/1143122297