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

Abstract

In 1983, N. Herrndorf proved that for a $\varphi$-mixing sequence satisfying the central limit theorem and ${\mathrm{liminf}}_{n\to \infty }\frac{{\sigma }_n^2}{n}>0$, 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.

12 pages

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