On the limiting distribution of sample central moments

TL;DR

This paper studies the asymptotic behavior of sample central moments, especially for singular distributions with limited support, and characterizes normality via asymptotic independence of sample mean and moments.

Contribution

It characterizes singular distributions with limited support and links their moments' limiting distribution to chi-square variables, also providing a new normality characterization.

Findings

Singular distributions have at most three support points.

Limiting distribution of moments can be expressed via chi-square variables.

Normality characterized by asymptotic independence of mean and moments.

Abstract

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called \emph{singular}, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the \emph{delta}-method, we show that the (second order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.