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• Author or Editor: U. Stadtmüller
Clear All Modify Search  # Kernel approximations of a wiener process

Periodica Mathematica Hungarica

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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# One-sided Tauberian conditions and double sequences

Periodica Mathematica Hungarica

## 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 .

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# Asymptotic properties of nonparametric curve estimates

Periodica Mathematica Hungarica
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# On the strong law of large numbers for delayed sums and random fields

Acta Mathematica Hungarica
Authors: A. Gut and U. Stadtmüller

## Abstract

A paper by Chow  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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# Тауберовы теоремы дря вэвещенных средних двоиных последователяностеи

Analysis Mathematica
Authors: U. Stadtmüller and У. Щтадтмуллер

## 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  as special case and generalize theorems given by Móricz  thereby showing that one of his Tauberian conditions can be weakened.

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