Strong approximation of vector-valued stochastic integrals (Q1181106)

For a \(p\)-dimensional \(L\)-mixing process \(\{f_ t,\;t\geq 0\}\) such that \(E(\int_ 0^ T| f_ t|^ 2 dt)<\infty\), for all \(T>0\), and \(E f_ tf^*_ t=S\) a.s., for all \(t\geq 0\), where \(f^*_ t\) denotes the transpose of \(f_ t\) and \(S\) is a constant \(p\times p\) matrix, it is proved that for any \(\varepsilon>0\), \(\int_ 0^ T f_ t dw_ t=\xi_ T+O_ M(T^{1/4+\varepsilon})\), where \(\{w_ t,\;t\geq 0\}\) is a real-valued standard Wiener process, \(\xi_ T\sim{\mathcal N}(0,TS)\). The symbol \(O_ M(T^{1/4+\varepsilon})\) denotes that the process \((\int_ 0^ T f_ t dw_ t-\xi_ T)T^{-1/4-\varepsilon}\) is \(M\)-bounded.