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

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.

### MSC:

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

Zbl 0790.60037
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