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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Riemann $ R\sb{1}$-summability of independent, identically distributed random variables

Author: Jack Cuzick
Journal: Proc. Amer. Math. Soc. 83 (1981), 119-124
MSC: Primary 40G99; Secondary 60G50
MathSciNet review: 619995
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Abstract: Let $ X,{X_1},{X_2}, \ldots $ be i.i.d. random variables. It is shown that $ E\left\vert X \right\vert{\log ^ + }{\log ^ + }\left\vert X \right\vert < \infty $ is a sufficient condition for Riemann $ {R_1}$-summability of $ \left\{ {{X_n}} \right\}$ to $ EX$. Counterexamples are provided which indicate that the strongest possible necessary condition of moment type is $ E\left\vert X \right\vert < \infty $. However under weak regularity conditions on the tails of the distribution of $ X$ the sufficient condition is also shown to be necessary.

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Keywords: Riemann summability, random Fourier series, almost sure convergence
Article copyright: © Copyright 1981 American Mathematical Society