A central limit theorem for stationary linear processes generated by associated process (Q2758203)

Let \(\varepsilon(t)\), \(t{\i}Z\), be a strictly stationary associated process, i.e. NEWLINE\[NEWLINE\text{cov}(f(\varepsilon(t_1),\dots, \varepsilon(t_m)), g(\varepsilon(t_1),\dots, \varepsilon(t_m)))\geq 0NEWLINE\]NEWLINE for \(f\) and \(g: R_m\to R\) increasing in each variable. Let \(E\varepsilon(t)= 0\), \(E\varepsilon^2(t)<\infty\) and \(0< \sigma^2= E\varepsilon^2(1)+ 2\sum_{t> 1} E(\varepsilon(1) \varepsilon(t))< \infty\). Put \(X(t)= \sum a_j\varepsilon(t- j)\), where \(\sum|a_j|< \infty\), and \(S(n)= X(1)+\cdots+ X(n)\). The following functional limit theorem is proved. The process \(\xi_n(u)\), \(0\leq u\leq 1\), with \(\sigma n^{1/2}(\sum a_j) S([nu])\) converges weakly to the Wiener process on \([0,1]\). Proofs start with the nonfunctional central limit theorem and \(n^{-1/2}\max\{|U(k)- S(k)|:k\leq n\}\to 0\) in probability, where \(U(k)= (\sum a_j)(\varepsilon(1)+\cdots+ \varepsilon(k))\).