ON THE ASYMPTOTIC BEHAVIOR OF WEAKLY LACUNARY SERIES

Let f be a measurable function satisfying f (x + 1) = f (x), integral(1)(0) f (x) dx = 0, Var([0,1])f < +infinity, and let (n(k))(k >= 1) be a sequence of integers satisfying n(k+1)/n(k) >= q > 1 (k = 1, 2 ,...). By the classical theory of lacunary series, under suitable Diophantine conditions on n(k), (f(n(k)x))(k >= 1) satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences (n(k))(k >= 1) as well, but as Fukuyama showed, the behavior of f(n(k)x) is generally not permutation-invariant; e.g. a rearrangement of the sequence can ruin the CLT and LIL. In this paper we construct an infinite order Diophantine condition implying the permutation-invariant CLT and LIL without any growth conditions on (n(k))(k >= 1) and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if f(x) = sin 2 pi x and (n(k))(k >= 1) grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences (n(k))(k >= 1) growing faster than polynomiallY, f(n(k)x))(k >= 1) has permutation-invariant behavior.