Kernel approximations of a Wiener process (Q1089993)

Consider a standard Wiener process W(.) on the real line and a kernel approximation of this process. The purpose of this paper is to study the deviation of these two processes, i.e. \[ X_ h(x)=h^{-1}\int K((v- x)/h)W(v)dv-W(x),\quad as\quad h\to 0+. \] Local and global weak and strong limit theorems for \(X_ h(.)\), \(h\to 0+\), are given, e.g. it is proven that for a certain class of kernels \[ \lim _{h\to 0+}\sup _{0\leq x\leq 1}X_ h(x)(2h \log h^{-1})^{-}=c\quad a.s. \] where \(c=c(K)\) is an explicit constant. Processes which correspond to \(X_ h\) occur in the asymptotics of nonparametric curve estimates.