A strictly stationary $\beta$-mixing process satisfying the central limit theorem but not the weak invariance principle

TL;DR

This paper constructs a strictly stationary $eta$-mixing process with finite moments where the central limit theorem holds but the weak invariance principle does not, highlighting a separation in these probabilistic properties.

Contribution

It provides a counterexample showing that the central limit theorem does not imply the weak invariance principle for certain mixing sequences.

Findings

Constructed a $eta$-mixing process with finite moments where CLT holds but WIP fails.

Demonstrated the separation between CLT and WIP in stationary sequences.

Extended understanding of mixing conditions and their implications in probability theory.

Abstract

In 1983, N. Herrndorf proved that for a $\phi$-mixing sequence satisfying the central limit theorem and $\liminf_{n\to\infty}\frac{\sigma^2_n}n>0$, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type ($\alpha$, $\beta$, $\rho$) of mixing the central limit theorem implies the weak invariance principle remained open. We construct a strictly stationary $\beta$-mixing sequence with finite moments of any order and linear variance for which the central limit theorem takes place but not the weak invariance principle.