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

Abstract

In 1983, N. Herrndorf proved that for a ϕ-mixing sequence satisfying the central limit theorem and lim infn→∞σn2/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.We construct 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.