Abstract
We prove that if Ik are disjoint blocks of positive integers and nk are independent random variables on some probability space (Ω,F,P) such that nk is uniformly distributed on Ik, then has, with P-probability 1, a mixed Gaussian limit distribution relative to the probability space ((0, 1),B, λ), where B is the Borel σ-algebra and λ is the Lebesgue measure. We also investigate the case when nk have continuous uniform distribution on disjoint intervals Ik on the positive axis.
Highlights
Salem and Zygmund [7] proved that if is a sequence of positive integers satisfying the Hadamard gap condition nk+1/nk ≥ q > 1 (k = 1, 2, . . .) (1.1) the sequence sin 2πnkx, k ≥ 1 obeys the central limit theorem, i.e.N −1/2 sin 2πnkx −→d N (0, 1/2) k=1 (1.2)with respect the the probability space (0, 1) equipped with Borel sets and Lebesgue measure
Berkes [1] showed that if N = ∪∞ k=1Ik where I1, I2, . . . are disjoint intervals of positive integers such that |Ik| → ∞, and n1, n2, . . . are independent random variables on some probability space (Ω, F, P) such that nk is uniformly distributed on Ik, with P-probability 1, sin 2πnkx satisfies the CLT (1.2)
In [1] the question was raised if the CLT (1.2) can hold for any sequence with nk+1 − nk = O(1)
Summary
Be a sequence of independent random variables on a probability space (Ω, A, P) such that nk is uniformly distributed over Ik. Let λk(x) = E(sin 2πnkx). Formula (1.4) shows that for any 0 < x < 1 we have limx→∞ g(x) = 1/2 and for large d the sequence sin 2πnkx − λk(x) nearly satisfies the ordinary CLT and LIL with limit distribution N (0, 1/2) and limsup = 1/2, just as lacunary trigonometric series with exponential gaps. This is not surprising since for large d the expected gaps E(nk+1 − nk) in our sequence are large.
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