On a Problem of Kac concerning Anisotropic Lacunary Sums
Abstract
Given a lacunary sequence (nk)k ∈ N, arbitrary positive weights (ck)k ∈ N that satisfy a Lindeberg-Feller condition, and a function f: T R whose Fourier coefficients fk decay at rate 1k1/2 + , we prove central limit theorems for Σk ≤ Nckf(nkx), provided (nk)k ∈ N satisfies a Diophantine condition that is necessary in general. This addresses a question raised by M. Kac [Ann. of Math., 1946].
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