Natural exponential families of probability distributions and exponential-polynomial approximation

Abstract

We consider expansions of regular functions in series with respect to infinite systems of Dirichlet polynomials, i.e., combinations of polynomials and exponents. We show that for every natural exponential family of probability measures on a real line and an infinite sequence of complex numbers, whose real parts belong to the natural parameter space of the above family, one can associate an expansion of any sufficiently regular function with respect to certain Dirichlet polynomials. We give an explicit formula for the remainder term in this expansion. It turns out to be an expectation of convolution of one of the above Dirichlet polynomials with certain differential operator of the function. The Dirichlet polynomials in question are combinations of exponents and Appel polynomials generated by members of the exponential family of distributions. Dirichlet series and expansions with respect to Appel polynomials are particular cases of the above expansions.