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SETS OF UNIQUENESS OF SERIES OF STOCHASTICALLY INDEPENDENT FUNCTIONS

Published online by Cambridge University Press:  14 October 2002

Francisco J. Freniche
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain ([email protected]; [email protected])
Ricardo Ríos-Collantes-de-Terán
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain ([email protected]; [email protected])
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Abstract

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It is shown that, for every sequence $(f_n)$ of stochastically independent functions defined on $[0,1]$—of mean zero and variance one, uniformly bounded by $M$—if the series $\sum_{n=1}^\infty a_nf_n$ converges to some constant on a set of positive measure, then there are only finitely many non-null coefficients $a_n$, extending similar results by Stechkin and Ul’yanov on the Rademacher system. The best constant $C_M$ is computed such that for every such sequence $(f_n)$ any set of measure strictly less than $C_M$ is a set of uniqueness for $(f_n)$.

AMS 2000 Mathematics subject classification:Primary 42C25. Secondary 60G50

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2002