Convergence rates in the functional CLT for alpha-mixing triangular arrays

TL;DR

This paper establishes convergence rates in the functional CLT for alpha-mixing triangular arrays under broad conditions, including non-uniform variance growth, with applications to statistical estimators and dynamical systems.

Contribution

It provides the first known convergence rates in the functional CLT for non-uniformly mixing triangular arrays without assumptions on variance growth.

Findings

Achieved near-optimal convergence rates for the functional CLT.

Derived rates for the classical CLT and moderate deviations.

Established Rosenthal type inequalities for the arrays.

Abstract

We obtain convergence rates (in the Levi-Prokhorove metric) in the functional central limit theorem (CLT) for partial sums $S_n=\sum_{j=1}^{n}\xi_{j,n}$ of triangular arrays $\{\xi_{1,n},\xi_{2,n},...,\xi_{n,n}\}$ satisfying some mixing and moment conditions (which are not necessarily uniform in $n$). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations principle and some Rosenthal type inequalities. We will also discuss applications to some classes of local statistics (e.g. covariance estimators), as well as expanding non-stationary dynamical systems, which can be reduced to non-uniformly mixing triangular arrays by an approximation argument. The main novelty here is that our results are obtained without any assumptions about the growth rate of the variance of $S_n$. The result are obtained using a certain type of block decomposition, which, in a sense, reduces the problem to the case when the variance of $S_n$ is "not negligible" in comparison with the (new) number of summands.