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March 2013 The complete characterization of a.s. convergence of orthogonal series
Witold Bednorz
Ann. Probab. 41(2): 1055-1071 (March 2013). DOI: 10.1214/11-AOP712

Abstract

In this paper we prove the complete characterization of a.s. convergence of orthogonal series in terms of existence of a majorizing measure. It means that for a given $(a_{n})^{\infty}_{n=1}$, $a_{n}>0$, series $\sum^{\infty}_{n=1}a_{n}\varphi_{n}$ is a.e. convergent for each orthonormal sequence $(\varphi_{n})^{\infty}_{n=1}$ if and only if there exists a measure $m$ on

\[T=\{0\}\cup\Biggl\{\sum^{m}_{n=1}a_{n}^{2},m\geq 1\Biggr\}\]

such that

\[\sup_{t\in T}\int^{\sqrt{D(T)}}_{0}(m(B(t,r^{2})))^{-{1}/{2}}\,dr<\infty,\]

where $D(T)=\sup_{s,t\in T}|s-t|$ and $B(t,r)=\{s\in T : |s-t|\leq r\}$. The presented approach is based on weakly majorizing measures and a certain partitioning scheme.

Citation

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Witold Bednorz. "The complete characterization of a.s. convergence of orthogonal series." Ann. Probab. 41 (2) 1055 - 1071, March 2013. https://doi.org/10.1214/11-AOP712

Information

Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1329.60062
MathSciNet: MR3077535
Digital Object Identifier: 10.1214/11-AOP712

Subjects:
Primary: 60G17
Secondary: 40A30 , 60G07

Keywords: majorizing measures , orthogonal series , Sample path properties

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • March 2013
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