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

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