Abstract
By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) n≥1 with 2π-periodic \({f\in L^2}\) this phenomenon is generally not valid and the asymptotic behavior of ( f (n k x)) k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n k . By the classical theory, the central limit theorem holds for f (n k x) if n k = 2k, or if n k+1/n k → α with a transcendental α, but it fails e.g., for n k = 2k − 1. The purpose of our paper is to give a necessary and sufficient condition for f (n k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened “infinitesimally”.
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C. Aistleitner was research supported by FWF grant S9603-N13.
I. Berkes was research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961.
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Aistleitner, C., Berkes, I. On the central limit theorem for f (n k x). Probab. Theory Relat. Fields 146, 267 (2010). https://doi.org/10.1007/s00440-008-0190-6
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DOI: https://doi.org/10.1007/s00440-008-0190-6