## Natural real exponential families with cubic variance functions.(English)Zbl 0714.62010

For a measure $$\mu$$ on the real line $${\mathbb{R}}$$ let P($$\theta$$) denote the probability distribution with $$\mu$$-density c exp($$\theta$$ x), $$x\in {\mathbb{R}}$$. The set of all distributions which arise in this way is the natural exponential family generated by $$\mu$$. This family can be parametrized by the mean m, the variance function V is the variance of P($$\theta$$) as a function of m. A number of well-known families of distributions arise in this way and have V being a polynomial. In this paper all families are characterized for which V is a polynomial of degree three, extending earlier work where this problem has been solved for polynomials up to degree two.
An interesting concept of reciprocity is introduced and exploited. It is shown that, in certain cases, reciprocity has an interpretation in terms of notions from fluctuation theory. As a result, some of the families can be identified as first entrance time distributions associated with certain Lévy processes.
Reviewer: R.Grübel

### MSC:

 62E10 Characterization and structure theory of statistical distributions 60E05 Probability distributions: general theory 60J99 Markov processes
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