On strong invariance principles under dependence assumptions (Q1074219)

Let \((x_ k)_{k\geq 1}\) be a sequence of mean zero \(R^ d\)-valued random variables. Denote \(S_ n(m)=x_{m+1}+...+x_{m+n}.\) Let \({\mathfrak F}_ m\) be the \(\sigma\)-algebra generated by \(x_ 1,...,x_ m\). Assuming that for some \(\theta >0\) \[ \| E[S_ n(m)| {\mathfrak F}_ m]\|_ 1\leq n^{-\theta}\quad uniformly\quad in\quad m \] or that for some constant C \((>0)\) and for all m,n (\(\geq 1)\) \[ \| E[S_ n(m)| {\mathfrak F}_ m]\|_ 2\leq C, \] the author proved strong invariance principles with order of approximation \(0(t^{-\kappa})\). The above dependence assumptions include various generalizations of martingales.