Critical LIL behavior of the trigonometric system

Abstract

It is a classical fact that for rapidly increasing ( n k ) ({n_k}) the sequence ( cos ⁡ n k x ) (\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 ) ({n_k}) grows faster than e c k {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 ∼ e c k {n_k} \sim {e^{c\sqrt k }} (critical behavior) and what is the probabilistic nature of ( cos ⁡ n k x ) (\cos {n_k}x) in the strongly dependent domain. In our paper we study the critical LIL behavior of ( cos ⁡ n k x ) (\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 ∼ e c k {n_k} \sim {e^{c\sqrt k }} .