On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

TL;DR

This paper establishes a rate of convergence in the weak invariance principle for dependent random variables under a new exponential mixing condition, with applications to Markov chains and spectral analysis.

Contribution

It introduces a novel mixing condition controlling dependence via characteristic functions, applicable to non-stationary processes and Markov chains.

Findings

Proves an invariance principle with explicit convergence rates.

Defines a new exponential mixing condition based on spectral properties.

Provides explicit bounds and constants for Markov chain applications.

Abstract

We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite dimensional increments of the process. The distinct feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited for processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.