The Central Limit Theorem for Some Weakly Dependent Sequences

Number Theory – Diophantine Problems, Uniform Distribution and Applications

2017

The theory of lacunary series starts with Weierstrass' famous example (1872) of a continuous, nondifferentiable function and now we have a wide and nearly complete theory of lacunary subsequences of classical orthogonal systems, as well as asymptotic results of thin subsequences of general function systems. However, many applications of lacunary series in harmonic analysis, orthogonal function theory, Banach space theory, etc. require uniform limit theorems for such series, i.e. theorems holding simultaneously for a class of lacunary series, and such results are much harder to prove than dealing with individual series. The purpose of this paper is to give a survey of uniformity theory of lacunary series and discuss new results in the field. In particular, we study the permutation-invariance of lacunary series and their connection with Diophantine equations, uniform limit theorems in Banach space theory, resonance phenomena for lacunary series and lacunary series with random gaps.

“…Gaposhkin [33] proved that f (n k x) obeys the CLT if n k+1 /n k → α where α r is irrational for r = 1, 2 . .…”

Section: Permutation-invariancementioning

confidence: 99%

“…Condition D 2 was introduced by Gaposhkin [33], who proved that under minor smoothness assumptions on f , it implies the CLT for f (n k x). Aistleitner and Berkes [1] proved that the CLT holds for…”

Section: Condition Dmentioning

confidence: 99%

On the Uniform Theory of Lacunary Series

Number Theory – Diophantine Problems, Uniform Distribution and Applications

2017

“…Gaposhkin also showed (see [12]) that the asymptotic behavior of N k=1 f (n k x) is intimately connected with the number of solutions of the Diophantine equation an k + bn l = c, 1 ≤ k, l ≤ N.…”

Section: Introductionmentioning

confidence: 96%

On the asymptotic behavior of weakly lacunary series

Aistleitner

Tichy

2011

Proc. Amer. Math. Soc.

Abstract. Let f be a measurable function satisfyingand let (n k ) k≥1 be a sequence of integers satisfying n k+1 /n k ≥ q > 1 (k = 1, 2, . . .). By the classical theory of lacunary series, under suitable Diophantine conditions on n k , (f (n k x)) k≥1 satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences (n k ) k≥1 as well, but as Fukuyama showed, the behavior of f (n k x) is generally not permutation-invariant; e.g. a rearrangement of the sequence can ruin the CLT and LIL. In this paper we construct an infinite order Diophantine condition implying the permutation-invariant CLT and LIL without any growth conditions on (n k ) k≥1 and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if f (x) = sin 2πx and (n k ) k≥1 grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences (n k ) k≥1 growing faster than polynomially, (f (n k x)) k≥1 has permutation-invariant behavior.

“…(1.11) Takahashi [23] proved that the CLT (1.9) also holds if n k+1 /n k → ∞ and f ∈ Lip (α), α > 0. The explanation of these phenomena was given in a profound paper of Gaposhkin [13], who showed the following remarkable result:…”

Section: Introductionmentioning

confidence: 99%

On the central limit theorem for f (n k x)

Aistleitner

Probab. Theory Relat. Fields

2008

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 ∈ 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 = 2 k , or if n k+1 /n k → α with a transcendental α, but it fails e.g., for n k = 2 k − 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".