A remark on a theorem of Berkes

Abstract

Let x be a standard Wiener process on \([0,\ 1]\), starting at the origin and with all its sample functions lying in the H\(\ddot{\text {o}}\)lder space \(H_{\alpha }^0\) of exponent \(\alpha ,\ 0< \alpha < \frac{1}{2}\). Define for \(n = 1, 2, \ldots \) processes \(x_n\) and \(z_n :\ x_n(t) = \frac{x(nt)}{\sqrt{n}}\) and \(z_n(t) = \frac{x(nt)}{\sqrt{n}f(n)},\ t \in [0,\ 1]\) where \(f(n) \rightarrow \infty \) and \(\frac{f(n)}{\sqrt{\log \log n}} \rightarrow 0\). In this paper, we show that the limit set of each of the random sequences \((x_n)\) and (\(z_n\)) of elements in \(H_{\alpha }^0\) in its norm topology is all of the space \(H_{\alpha }^0\).