On the functional CLT for slowly mixing triangular arrays

TL;DR

This paper develops new conditions for the functional CLT in triangular arrays with slow mixing, relaxing variance assumptions and using maximal moment and mixing rate conditions, applicable to $ ho$- and $eta$-mixing arrays.

Contribution

It introduces alternative conditions for the functional CLT that do not rely on variance sum assumptions, focusing on maximal moments and specific mixing rate criteria.

Findings

Established functional CLT under new mixing rate conditions

Extended applicability to $ ho$- and $eta$-mixing arrays

Relaxed variance assumptions compared to prior work

Abstract

In \cite{MPU} a functional CLT was obtained for triangular arrays satisfying the Lindeberg condition, that the sum of the individual variances is at most the same order as the variance of the underlying sum, and under the optimal mixing rats $\sum_{n}\rho(2^n)<\infty$, where $\rho(\cdot)$ are the $\rho$-mixing coefficients of the array. In this paper we will present alternative conditions which do not involve the assumption on the sum of variances, and instead we will assume certain maximal moment assumptions (which we can verify for $\phi$-mixing arrays) and mixing rates of the form $\sum_n\rho(e^{G(n)})<\infty$ where $G(n)$ grows sub-linearly fast in $n$ (e.g. $G(n)=n/\ln(\ln n)$). We will also discuss alternative conditions to the ones in the functional CLT for $\alpha$-mixing triangular arrays which was obtained in \cite{MP}.