Let and be two integrable real-valued random variables and let be the quantile function of . Then if is integrable over we have

where is the a-mixing coefficient.

* Proof:* Set and then

since and

note also that

which implies that

Now since

Now let be a r.v. with uniform distribution over and let be a bivariate r.v. defined as

Hence

and

thus

**References**:

D.Bosq (1996). Nonparametric Statistics for Stochastic Processes. Springer

E.Rio (1993). *Covariance inequalities* for *strongly mixing processes*. Annales de l’institut Henri Poincaré (B) Probabilités *et Statistiques* 29.4 : 587-597.

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