Characterization of quadratic Cauchy-Stieltjes kernel families based on the orthogonality of polynomials
Many classifications of natural exponential families (NEF) are based on the form of their variance functions which take a very simple form for many common NEFs. A result in [\textit{J. Meixner}, J. Lond. Math. Soc. 9, 6--13 (1934; JFM 60.0293.01)] characterizes the distribution \(\mu\) for which a family of \(\mu\)-orthogonal polynomials exists with an exponential generating function. These distributions generate exactly the class of NEFs from [\textit{C. N. Morris}, Ann. Stat. 10, 65--80 (1982; Zbl 0498.62015)] with the property that the variance function is a polynomial in the mean of degree less than or equal to two. Another characterization in [\textit{P. Feinsilver}, Proc. Am. Math. Soc. 98, 298--302 (1986; Zbl 0615.60050)] shows that a certain class of polynomials naturally associated with a NEF is \(\mu\)-orthogonal if and only if the family is in the Morris class. The paper under review studies the class of quadratic families related with the Cauchy-Stieltjes kernel (CSK) \(1/(1-\theta x)\) instead with the exponential kernel \(\exp(\theta x)\), in the spirit of the results of Meixner and Feinsilver. The main result gives the characterization of the sequence of polynomials corresponding to the distribution generating a quadratic CSK family based on the property of orthogonality. Then the author determines the families of orthogonal polynomials with a Cauchy-Stieltjes type generating function, which leads to another characterization of the quadratic CSK family.
scientific article; zbMATH DE number 6817560