Random Fourier transforms

Paley and Zygmund [10](1) have shown that for almost every random choice of signs the series represents a continuous function, provided only that X fln(lg n) < 00 .A little later B. Jessen [5] studied random almost periodic functions.One of his theorems, stated in the language of probability, is this: Let Xi, X2, • • • be real and let Yi, Yi, ■ ■ ■ be independent complex random variables, each Yn being uniformly distributed on the circumference \ =1.If Z2 <r|X"| ane < «j for some <r>0, then almost certainly £ a"FneiX»< -o(lg i)1'*, t-*<*>.It is our purpose to show that the ±an or the a"Yn may be replaced by any independent random variables Xi, X2, • • • subject to £{J"| =0 and 2^£{A^n} < °°-(Here £{-Xj denotes the expectation of X.) We shall define Z(t) = £ Xne**' and prove that if Z£{xl\{k(i + \)}1+€<™, then almost certainly Z(t) is continuous in t; that if T,£{xl}\2a< oo for some positive a^l, then almost certainly z(t) = o('2, *->«, and moreover Z(t) has modulus of continuity of the form K~ha(lg 1/h)112.More precise statements will be found in Theorems 2 ( §1), 4 ( §5), 7 ( §8).The proofs are based on two inequalities which are of interest in themselves.They are Lemma 1 ( §2) and Lemma 10 ( §6).Luckily both can be ex-