A quenched weak invariance principle

TL;DR

This paper establishes a quenched weak invariance principle, proving a functional form of the almost sure conditional central limit theorem for certain dependent random variables, with applications to mixing processes and Markov chains.

Contribution

It introduces a new approach using normal approximation of double indexed martingale-like sequences to prove the invariance principle under projective criteria.

Findings

Proves a quenched weak invariance principle for dependent variables.

Applies results to strongly mixing processes.

Extends to non-irreducible Markov chains.

Abstract

In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and non irreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, a theory which has interest in itself.