Quenched central limit theorems for a stationary linear process

TL;DR

This paper establishes conditions under which a stationary linear process satisfies a quenched central limit theorem, highlighting cases where classical CLT results do not hold uniformly across different realizations.

Contribution

It identifies a sufficient condition for quenched CLT in stationary linear processes and provides examples where the CLT fails to be quenched despite satisfying other conditions.

Findings

A stationary linear process satisfying Maxwell-Woodroofe condition can have non-quenched CLT.

The variance of partial sums can be o(n) while still satisfying the CLT.

Weak invariance principle may fail even when CLT holds.

Abstract

We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are o(n), there is a CLT with a convergence towards N(0,1) when dividing by standard deviation of the partial sums, and the CLT is not quenched. The weak invariance principle does not hold.