Studia Scientiarium Mathematicarum Hungarica 34. (1998)
1-3. szám - Berkes I.-Philipp W.: A limit theorem for lacunary series ?f(nkx)
Studio. Scientiamin Mathematicarum Hungáriái ,V, (1998), 1 13 A LIMIT THEOREM FOR LACUNARY SERIES £ f(nkx) I. BERKES* 1 and W. PHILIPP To the memory of Alfréd Rényi Abstract Let f: R—t R be a Lebesgue measurable function satisfying Several authors investigated the asymptotic properties of lacunary series y\ckf(nkx) under the Hadamard gap condition ' A+t(fe = 1,2,...) and the behaviour of such series is well known. On the other hand, very little is known on the properties of ^2 ck f(n/.x) if («<•) grows slower than exponentially. The purpose of this paper is to prove an asymptotic result for such series. 1. Introduction Let /: R —> R be a Lebesgue measurable function satisfying The asymptotic properties of lacunary series Y2ckf{'nkx) have been investigated by many authors and are known to be very similar to those of independent random variables. For example, Takahashi proved ([13], [14]) that if / is a Lipsehitz function satisfying (1.1) and (nk) is a sequence of positive integers satisfying (1.2) nk+l/7ik —> oo 1991 Mathematics Subject Classification. Primary 42A55, 60F15. Key words and phrases. Lacunary series, law of the iterated logarithm. 1 Research supported by Hungarian National Foundation for Scientific Research Grants T 16384 and T 19346. 0081-6906/98/$ 5.00 ©1998 Akadémiai Kiadó, Budapest MAGYAR •TUDOMÁNYOS AKADÉMIA l l /(x + l) = /(x), j f(x)dx = 0, I f2(x)dx = 1. o o l l (1.1) f(x + l) = /(x), J f{x)dx = 0, I f2(x)dx= 1. o b