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

In 1983, N. Herrndorf proved that for a ϕ -mixing sequence satisfying the central limit theorem and lim inf n → ∞ σ 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 ( α , β , ρ ) of mixing the central limit theorem implies the weak invariance principle remained open. struct a strictly stationary β -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.