Random Fourier transforms
G. A. Hunt · Transactions of the American Mathematical Society · 1951
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) .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} «, 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-