On degenerate sums of $m$-dependent variables

TL;DR

This paper investigates when the central limit theorem for sums of m-dependent variables results in a non-degenerate limit, providing a criterion based on the structure of the variables and applications to random trees.

Contribution

It characterizes the conditions under which the CLT limit is non-degenerate for m-dependent variables and extends the result to block factors.

Findings

Degenerate limits occur only when variables are differences of an (m-1)-dependent sequence.

Provides a simple criterion to determine non-degeneracy of the CLT limit.

Includes applications to subtree counts in random trees.

Abstract

It is well-known that the central limit theorem holds for partial sums of a stationary sequence $(X_i)$ of $m$-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if $\mathrm{Var}(X_i)\neq0$. We show that this happens only in the case when $X_i-\mathbb E X_i=Y_i-Y_{i-1}$ for an $(m-1)$-dependent stationary sequence $(Y_i)$ with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to degenerate. Two applications to subtree counts in random trees are given.