A note on mixing properties of certain associated processes (Q1307398)

The paper deals with integer-valued stochastic processes \((X_n)_{n\in N}\) called associated processes, and with strongly mixing stochastic processes. The aim is to prove mixing properties and an invariance principle for a class of associated processes. A process is strongly mixing if the so-called mixing coefficient \(\alpha(n)\) (defined in terms of the dependence between events distant in time) goes to zero as \(n\to\infty\). The basic lemma states that for an integer-valued associated square integrable process the mixing coefficient \(\alpha(n)\) can be estimated in terms of a certain covariance coefficient \(u(n)\): the inequality \(\alpha(n) \leq 4 \sum^\infty_{i=0}u(n+i)\) holds for every \(n\in N\). Here \(u(n)=\sup_{k\in N}\sum_{|j-k|\geq n}\text{Cov} (X_j,X_k)\). It follows that an integer-valued associated square integrable process is strongly mixing if \(u(n)=O(n^{-\lambda})\) for some \(\lambda >1\), and the decay rate of \(\alpha(n)\) can be estimated in terms of the decay of \(u(n)\). Moreover, certain limit theorems for strongly mixing processes can now be applied to associated processes. In particular, the author gives a central limit theorem and an invariance principle for strictly stationary integer-valued associated square integrable processes with \(u(n)=O(n^{-\lambda})\) for some \(\lambda>1\) (under additional integrability assumptions).