A limit theorem for lacunary series Sigma f(n(k)x)

Let f: R --> R be a Lebesgue measurable function satisfying f(x + 1) = f(x), integral(0)(1) f(x)dx = 0, integral(0)(1) f(2)(x)dx = 1. Several authors investigated the asymptotic properties of lacunary series S igma c(k)f(n(k)x) under the Hadamard gap condition n(k+1)/n(k) greater than or equal to q > 1 (k = 1, 2,...) and the behaviour of such series is well known. On the other hand, very lit tle is known on the properties of Sigma c(k)f(n(k)x) if (n(k)) grows slower than exponentially. The purpose: of this paper is to prove an asymptotic r esult for such series.