Lacunary series and probability
Robert Bruce Kaufman · Pacific Journal of Mathematics · 1971
In this note we continue some investigations connecting a lacunary series A of real numbersand a probability measure μ on (-°o, °°) satisfying (1) μ&a, a + A]) « hfi for all intervals [a, a + h] of length h < 1, and a fixed exponent 0 < β < 1. (The notation X « F is a substitute for X = 0(Y).)Measures jM occur in the theory of sets of fractional Hausdorff dimension.In the following statements S is a subset of (-oo, oo) of Lebesgue measure 0, depending only on μ and A. THEOREM 1.For r = 2, 4, 6, and t&S, there is a constant B r (t) so that s: I 2 α* cos (λ k tx + h) \ r μ(dx) S Here B r (f) is independent of the sequences (aj) and (bk).THEOREM 2. For £ g S the normalized sums tend in law (with respect to the probability μ) to the normal law.Here the convergence is uniform for all sequences (bk). 1 J-ooJ-ooJl inf (1, 2\yx,yx 2 \~1)μ(dx 1 )μ(dx 2 ) .o