# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Natural real exponential families with cubic variance functions. (English) Zbl 0714.62010
For a measure $\mu$ on the real line ${\bbfR}$ let P($\theta$) denote the probability distribution with $\mu$-density c exp($\theta$ x), $x\in {\bbfR}$. 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 General theory of probability distributions 60J99 Markov processes
Full Text: