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When the cone condition fails

Analysis Mathematica
Volume/Issue:
Volume 31: Issue 4
DOI:
https://doi.org/10.1007/s10476-005-0020-3
Author: Michel Weber
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Controlling Orthogonal Series with Irrational Rotations

Periodica Mathematica Hungarica
Volume/Issue:
Volume 38: Issue 1-2
DOI:
https://doi.org/10.1023/a:1004771704266
Author: Michel Weber
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Divisors, spin sums and the functional equation of the Zeta-Riemann function

Periodica Mathematica Hungarica
Volume/Issue:
Volume 51: Issue 1
DOI:
https://doi.org/10.1007/s10998-005-0024-6
Author: Michel Weber

Summary  

Let \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} $S_n$ \end{document}, \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} $n=1,2\dots$ \end{document} be the sequence of partial sums of independent spin random variables. We show that the distribution value of the divisors 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} $S_n$ \end{document}, is intimately related to the Zeta-Riemann function via the functional equation and Theta elliptic functions.

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On a  stronger form  of Salem-Zygmund's inequality for random trigonometric sums with examples

Periodica Mathematica Hungarica
Volume/Issue:
Volume 52: Issue 2
DOI:
https://doi.org/10.1007/s10998-006-0013-4
Author: Michel Weber

Summary  

By applying the majorizing measure method, we obtain a new estimate  of the supremum of random  trigonometric sums. We show that this estimate is strictly stronger than the well-known Salem-Zygmund's estimate, as well as recent general formulations of it obtained by the author. This improvement is obtained by considering the case when the characters are indexed on  sub-exponentially growing sequences of integers. Several remarkable examples are studied.

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О тождестве Ки Фана

Analysis Mathematica
Volume/Issue:
Volume 34: Issue 3
DOI:
https://doi.org/10.1007/s10476-008-0304-5
Author: Michel Weber

Abstract  

We give several applications of an identity for sums of weakly stationary sequences due to Ky Fan.

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On the strong law of large numbers and additive functions

Periodica Mathematica Hungarica
Volume/Issue:
Volume 62: Issue 1
DOI:
https://doi.org/10.1007/s10998-011-5001-7
Authors: István Berkes, Wolfgang Müller, and Michel Weber

Abstract  

Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X 1,X 2, … is any sequence of integrable i.i.d. random variables, then

\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} $$\mathop {\lim }\limits_{N \to \infty } \frac{{\sum\nolimits_{n = 1}^N {f(n)X_n } }} {{\sum\nolimits_{n = 1}^N {f(n)} }} = \mathbb{E}Xa.s.$$ \end{document}

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