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 the sequence behaves like a sequence of i.i.d. random variables. Actually, this almost i.i.d. behavior holds if grows faster than ; 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 (critical behavior) and what is the probabilistic nature of in the strongly dependent domain. In our paper we study the critical LIL behavior of 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 .

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1993-1099352-2

Keywords:
Lacunary trigonometric series,
weak and strong dependence,
law of the iterated logarithm,
upper-lower class tests

Article copyright:
© Copyright 1993
American Mathematical Society