Probability distributions with binomial moments

Abstract

We prove that if $p\ge 1$ and $-1\le r\le p-1$ then the binomial sequence $\left(\genfrac{}{}{0ex}{}{np+r}{n}\right)$, $n=0,1,...$, is positive definite and is the moment sequence of a probability measure $\nu (p,r)$, whose support is contained in $\left[0,p^p(p-1{)}^{1-p}\right]$. If $p>1$ is a rational number and $-1<r\le p-1$ then $\nu (p,r)$ is absolutely continuous and its density function $V_{p,r}$ can be expressed in terms of the Meijer $G$-function. In particular cases $V_{p,r}$ is an elementary function. We show that for $p>1$ the measures $\nu (p,-1)$ and $\nu (p,0)$ are certain free convolution powers of the Bernoulli distribution. Finally we prove that the binomial sequence $\left(\genfrac{}{}{0ex}{}{np+r}{n}\right)$ is positive definite if and only if either $p\ge 1$, $-1\le r\le p-1$ or $p\le 0$, $p-1\le r\le 0$. The measures corresponding to the latter case are reflections of the former ones.

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