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) 0, then almost certainly £ a"FneiX». 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^£{Aⁿ «, 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-