A random functional central limit theorem for stationary linear processes generated by martingales
Let \(X_t= \sum^\infty_{j=-\infty} a_j\varepsilon_{t-j}\), \(\sum^\infty_{j=-\infty}|a_j|<\infty\) be a stationary ergodic linear process defined on a probability space \((\Omega,{\mathbf F},P)\), where \(\{\varepsilon_t, t\in\mathbb{Z}\}\) is a strictly stationary sequence of martingale differences \[ E(\varepsilon_t\mid{\mathbf F}_{t-1})= 0,\quad E(\varepsilon^2_t\mid{\mathbf F}_{t-1})= \sigma^2<\infty,\quad\text{a.s.} \] Here, \({\mathbf F}_t\) is the sub-\(\sigma\)-algebra generated by \(\varepsilon_s\), \(s\leq t\). Let \[ S_n= \sum^n_{t=1} X_t,\quad \tau^2=\sigma^2\Biggl( \sum^\infty_{j=-\infty} a_j\Biggr)^2, \] \[ \xi_u(u)= {1\over\sqrt n\tau} (S_r+ (un-r) X_{r+1});\quad r/n\leq u\leq(r+1)/n\quad (0\leq r\leq n-1). \] The authors prove that all finite-dimensional distributions of \(\xi_n\) converge weakly under the probability measure \(P(\cdot\mid B)\) (\(B\in{\mathbf F}_k\), \(P(B)>0\), \(k\) being fixed arbitrary) to the finite-dimensional distributions of a Wiener process and that \(\{\xi_{N_n}(u): 0\leq u\leq 1\}\) converges weakly to a Wiener process where \(\{N_n\}\) is a sequence of positive integer-valued random variables defined on the probability space such that \(N_n/n\) converges in probability to a real-valued random variable \(N\) with \(P(0<N<\infty)=1\).
scientific article; zbMATH DE number 1092032