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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On random Fourier series


Authors: Jack Cuzick and Tze Leung Lai
Journal: Trans. Amer. Math. Soc. 261 (1980), 53-80
MSC: Primary 60G17; Secondary 42A20, 60F15
DOI: https://doi.org/10.1090/S0002-9947-1980-0576863-8
MathSciNet review: 576863
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Abstract: Motivated by Riemann's $ {R_1}$ summation method for i.i.d. random variables $ {X_1},\,{X_2},\, \ldots $, this paper studies random Fourier series of the form $ \sum\nolimits_1^\infty {{a_n}{X_n}\,\sin (nt\, + \,{\Phi _n})} $, where $ \{ {a_n}\} $ is a sequence of constants and $ \{ {\Phi _n}\} $ is a sequence of independent random variables which are independent of $ \{ {X_n}\} $. Questions of continuity and of unboundedness are analyzed through the interplay between the asymptotic properties of $ \{ {a_n}\} $ and the tail distribution of $ {X_1}$. A law of the iterated logarithm for the local behavior of the series is also obtained and extends the classical result for Brownian motion to a general class of random Fourier series.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0576863-8
Keywords: Riemann's $ {R_1}$ summability, convergence, continuity, unboundedness, law of the iterated logarithm, random Fourier series
Article copyright: © Copyright 1980 American Mathematical Society