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

Davide Giraudo (1) , Dalibor Volný (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

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.

Domaines: Probabilités [math.PR]

Davide Giraudo, Dalibor Volný. A strictly stationary $\beta$-mixing process satisfying the central limit theorem but not the weak invariance principle. Stochastic Processes and their Applications, 2014, 124, pp.3769-3781. ⟨10.1016/j.spa.2014.06.008⟩. ⟨hal-00911758v2⟩