Conventional central limit theory concerns the asymptotic normality of centered and scaled sums of i.i.d. random variables with finite variances. The corresponding enormous domain of attraction of the normal distribution is, in a certain sense, also its biggest liability, because no other feature of the distribution beyond its first two moments is used in the normal approximation. A more differentiated approach is obtained by considering so-called Tweedie asymptotics, which involve exponentially tilted and scaled sums of i.i.d. random variables with finite variances. The resulting set of limiting distributions form the three-parameter class of Tweedie exponential dispersion models with power variance functions. Tweedie asymptotics covers a considerable range of different types of asymptotics, ranging from Poisson and compound Poisson convergence via a gamma approximation to results involving exponentially tilted extreme stable distributions. We review the main results of Tweedie asymptotics in the setting of convergence of exponentially tilted and scaled Lévy processes.

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Keywords: Hougaard Lévy process; Power variance function; Fixed point of renormalization group; Exponential family of Lévy processes

Biography: Bent Jørgensen is Professor in the Department of Mathematics and Computer Science, University of Southern Denmark, where he has been since 1997. Before that he was at IMPA, Rio de Janeiro, Brazil and The University of Brithish Columbia, Vancouver, Canada. His main research areas are exponential dispersion models and statistical analysis of correlated data.