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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Critical LIL behavior of the trigonometric system


Author: I. Berkes
Journal: Trans. Amer. Math. Soc. 338 (1993), 553-585
MSC: Primary 60F15; Secondary 42A55
MathSciNet review: 1099352
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Abstract: It is a classical fact that for rapidly increasing $ ({n_k})$ the sequence $ (\cos {n_k}x)$ behaves like a sequence of i.i.d. random variables. Actually, this almost i.i.d. behavior holds if $ ({n_k})$ grows faster than $ {e^{c\sqrt k }}$; below this speed we have strong dependence. While there is a large literature dealing with the almost i.i.d. case, practically nothing is known on what happens at the critical speed $ {n_k} \sim {e^{c\sqrt k }}$ (critical behavior) and what is the probabilistic nature of $ (\cos {n_k}x)$ in the strongly dependent domain. In our paper we study the critical LIL behavior of $ (\cos {n_k}x)$ i.e., we investigate how classical fluctuational theorems like the law of the iterated logarithm and the Kolmogorov-Feller test turn to nonclassical laws in the immediate neighborhood of $ {n_k} \sim {e^{c\sqrt k }}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1099352-2
PII: S 0002-9947(1993)1099352-2
Keywords: Lacunary trigonometric series, weak and strong dependence, law of the iterated logarithm, upper-lower class tests
Article copyright: © Copyright 1993 American Mathematical Society