Vol. 36, No. 1, 1971

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Lacunary series and probability

Robert P. Kaufman

Vol. 36 (1971), No. 1, 195–200
Abstract

In this note we continue some investigations connecting a lacunary series Λ of real numbers

Λ : 1 ≦ λ1 < ⋅⋅⋅ < λk < ⋅⋅⋅ ,qλk ≦ λk+1 (1 < q)

and a probability measure μ on (−∞,) satisfying

μ([a,a + h]) ≪ hβ
(1)

for all intervals [a,a + h] of length h < 1, and a fixed exponent 0 < β < 1. (The notation X Y is a substitute for X = 0(Y ).) Measures μ occur in the theory of sets of fractional Hausdorff dimension.

In the following statements S is a subset of (−∞,) of Lebesgue measure 0, depending only on μ and Λ.

Theorem 1. For r = 2,4,6, and tS, there is a constant Br(t) so that

∫ ∞ ∑                               ∑
|  ak cos(λktx+ bk)|rμ(dx) ≦ Br (t)(  |ak|2)r∕2
−∞

Here Br(t) is independent of the sequences (aj) and (bk).

Theorem 2. For tS the normalized sums

 1  −1∕2 ∑
(2N)        cos(λktx+ bk)
k≦N

tend in law (with respect to the probability μ) to the normal law. Here the convergence is uniform for all sequences (bk).

Mathematical Subject Classification
Primary: 60.30
Milestones
Received: 21 February 1969
Published: 1 January 1971
Authors
Robert P. Kaufman