The Annals of Probability

The complete characterization of a.s. convergence of orthogonal series

Witold Bednorz

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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.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 1055-1071.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750950

Digital Object Identifier
doi:10.1214/11-AOP712

Mathematical Reviews number (MathSciNet)
MR3077535

Zentralblatt MATH identifier
1329.60062

Subjects
Primary: 60G17: Sample path properties
Secondary: 40A30: Convergence and divergence of series and sequences of functions 60G07: General theory of processes

Keywords
Sample path properties majorizing measures orthogonal series

Citation

Bednorz, Witold. The complete characterization of a.s. convergence of orthogonal series. Ann. Probab. 41 (2013), no. 2, 1055--1071. doi:10.1214/11-AOP712. https://projecteuclid.org/euclid.aop/1362750950


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References

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