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  • 1 Abteilung für Mathematik, I Universität Ulm Oberer Eselsberg D-7900 Ulm Federal Republic of Germany
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Let a standard Wiener processW(.) be given on the real line. We investigate the asymptotic behaviour of

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X_h (x) = h^{ - 1} \smallint K((\upsilon - x)h^{ - 1} )W(\upsilon )d\upsilon - W(x),$$ \end{document}
ash → 0 +, that is the deviation of a kernel approximation ofW(.) from the process itself. For example, we confirm, under certain conditions onK, a conjecture of P. Révész proving that
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim \sup }\limits_{h \to 0 + 0 \leqslant x \leqslant 1} \frac{{\left| {X_h (x)} \right|}}{{\sqrt {h \log h^{ - 1} } }} \underline{\underline {a.s.}} c,$$ \end{document}
with an explicit constantc = c(K).

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