On generating functions of Hausdorff moment sequences

TL;DR

This paper characterizes generating functions of Hausdorff moment sequences, linking them to Pick functions, and explores their properties for various probability distributions, providing new insights and proofs in the theory of moments.

Contribution

It offers new characterizations of generating functions for Hausdorff moments, including convex and concave distributions, and provides a simple proof for when Fuss-Catalan and Raney numbers are moments of probability distributions.

Findings

Generating functions are characterized as Pick functions positive on (-∞,1).

Fuss-Catalan and Raney numbers are moments iff p ≥ 1 and p ≥ r ≥ 0.

Provides new proofs and characterizations for moments of specific combinatorial sequences.

Abstract

The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-\infty,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$. Also we provide a simple analytic proof that for any real $p$ and $r$ with $p>0$, the Fuss-Catalan or Raney numbers $\frac{r}{pn+r}\binom{pn+r}{n}$, $n=0,1,\ldots$ are the moments of a probability distribution on some interval $[0,\tau]$ {if and only if} $p\ge1$ and $p\ge r\ge 0$. The same statement holds for the binomial coefficients $\binom{pn+r-1}n$, $n=0,1,\ldots$. A corrigendum (Trans. Amer. Math.Soc., to appear) has been included as an appendix, correcting gaps in the proof of Lemma 3.