On a problem of Csörgö and Révész
Let \(\{X_ n\}\) be a sequence of i.i.d. random variables with mean 0 and variance 1. Let \(S_ n=X_ 1+...+X_ n\) and \(0<r<1\). It is well known that, by redefining \(\{S_ n\}\) on a richer probability space if necessary, there exists a standard Wiener process \(\{\) W(t), \(t>0\}\) such that \[ S_ n-W(n)=O((\log n)^{1/r})\quad a.s. \] when E exp(t\({}_ 0| X_ 1|^ r)<\infty\) for some \(t_ 0>0\). Answering a question posed in \textit{M. Csörgö} and \textit{P. Révész} [Strong approximations in probability and statistics. (1981)] the author proves that O((log n)\({}^{1/r})\) cannot be replaced by o((log n)\({}^{1/r})\). He also gives two theorems on the increments of \(S_ n\), supplementing some theorems in the book cited above.
scientific article; zbMATH DE number 4121128