Covariance inequalities for strongly mixing processes. (English) Zbl 0798.60027

Rio, Emmanuel

Ann. Inst. Henri Poincaré, Probab. Stat. 29, No. 4, 587-597 (1993).

It is proved that \[ \bigl | \text{cov} (X,Y) \bigr | \leq 2 \int^{2\alpha}_ 0 Q_ x (u)Q_ y (u)du \] , where \(Q_ x(u) = \inf \{t : P(| x |>t) \leq u\}\) and \(\alpha\) is the strong mixing coefficient between two \(\sigma\)-fields generated respectively by real- valued random variables \(X\) and \(Y\). This inequality, extending e.g. Davydov’s one, is sharp, up to a constant factor. It is applied to obtain bounds of the variance of sums of strongly mixing processes and fields. In a recent paper by P. Doukhan, P. Massart and the author [ibid. 30, No. 1, 63-82 (1994; Zbl 0790.60037)] it is used also to prove the functional CLT for strongly mixing processes.

Reviewer: A.V.Bulinskij (Moskva)

MSC:

60F05	Central limit and other weak theorems
60F17	Functional limit theorems; invariance principles

Keywords:

covariance inequalities; strongly mixing processes; functional central limit theorem for strongly mixing processes

Citations:

Zbl 0790.60037