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  • Author or Editor: U. Stadtmüller x
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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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Abstract  

We give a theorem of Vijayaraghavan type for summability methods for double sequences, which allows a conclusion from boundedness in a mean and a one-sided Tauberian condition to the boundedness of the sequence itself. We apply the result to certain power series methods for double sequences improving a recent Tauberian result by S. Baron and the author [4].

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Abstract  

A paper by Chow [3] contains (i.a.) a strong law for delayed sums, such that the length of the edge of the nth window equals n α for 0 < α < 1. In this paper we consider the kind of intermediate case when edges grow like n=L(n), where L is slowly varying at infinity, thus at a higher rate than any power less than one, but not quite at a linear rate. The typical example one should have in mind is L(n) = log n. The main focus of the present paper is on random field versions of such strong laws.

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Abstract  

We discuss the relations between weighted mean methods and ordinary convergence for double sequences. In particular, we study Tauberian theorems also for methods not being products of the related one-dimensional summability methods. For the special case of theC 1,1-method, the results contain a classical Tauberian theorem by Knopp [9] as special case and generalize theorems given by Móricz [16] thereby showing that one of his Tauberian conditions can be weakened.

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