NON-GAUSSIAN LIMIT DISTRIBUTIONS OF LACUNARY TRIGONOMETRIC SERIES

It is a well known fact that for rapidly increasing n(k) the sequence (cos n(k)x)k = 1 infinity behaves like a sequence of independent random variables; in particular N-1/2-SIGMA-k less-than-or-equal-to N cos n(k)x has a limiting Gaussian distribution as N --> infinity. Under a certain critical speed (actually n(k) approximately e square-root k) this result breaks down and (cos n(k)x)k = 1 infinity becomes strongly dependent. The purpose of this paper is to investigate the asymptotic behavior of normed sums a(N)-1-SIGMA-k less-than-or-equal-to N cos n(k)x in the strongly dependent domain; specifically, we construct a large class of nongaussian limit distributions of such sums.