CONTENTSIntroductionChapter I. Lacunary subsystems of general systems of functions § 1. Definitions and notation § 2. Convergence § 3. Integrability § 4. Absolute convergence § 5. The central limit theorem § 6. The law of the iterated logarithmChapter II. Lacunary subsystems of specific systems of functions § 1. The trigonometric system § 2. The Walsh system § 3. Orthogonal polynomials § 4. Systems of the form References

Lacunary Series and Independent Functions

During the last three decades the concept “invariance principle” has undergone several subtle changes. In the early 1950’s an invariance principle was a result that nowadays often would be called a functional central limit theorem (FCLT). At present the term “invariance principle” generally stands as a synonym for an approximation theorem: A given process, such as a partial sum process, an empirical process, an extremal process, a U-statistic, etc. is approximated in distribution, in probability, in LP or almost surely by a canonical process, such as a Brownian motion, a Kiefer process, a special extremal process or in case of a U-statistic by a multiple stochastic integral.

Invariance Principles for Independent and Weakly Dependent Random Variables

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
  = 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”.

/pdf/on-the-central-limit-theorem-for-f-nkx-2yfgondlzx.pdf

On the central limit theorem for f (nkx)

Хорошо известно, что в ероятностное поведе ние лакунарного тригоно метрического ряда {cos 2πnkx} тесно связ ано с «критическим» у словием лакунарности

$$\frac{{n_{k + 1} }}{{n_k }} \geqq 1 + \frac{{c_k }}{{\sqrt k }},c_k \to \infty $$

(*)

. Например, если выполн ено условие (*), то последовательность {cos2πnkx} удовлетворяет центральной предель ной теореме, и при этом условие (*) не может быть ослабле но. Для последовательносте й, удовлетворяющих (*), и звестны и другие результаты по добного рода, в то время как для более медленно расту щих последовательносте й {nk} не известно, по-видимому, ничего. В с татье развит метод, ко торый при помощи мартингально й техники позволяет проводить исследование систем {cos 2πnkx} для последовательно стей, не удовлетворяю щих условию (*). Получено про стое объяснение условия (*), изучено, как «пропа-дает» центральная предель ная теорема при посте пенном ослаблении условия (*) и дока-заны некоторые центральн ые предельные теорем ы в отсутствие этого усл овия. Получены другие предельные те оремы для {cos 2πnkx}, напри мер, закон повторного лог арифма и принципы инвариантн ости.

On the central limit theorem for lacunary trigonometric series

/pdf/on-the-law-of-the-iterated-logarithm-for-lacunary-4w7ggs2o68.pdf

On the law of the iterated logarithm for lacunary trigonometric series, II

/pdf/the-law-of-the-iterated-logarithm-for-a-gap-sequence-with-3w93c61ohr.pdf

The law of the iterated logarithm for a gap sequence with infinite gaps

/pdf/a-gap-sequence-with-gaps-bigger-than-the-hadamard-s-n7zjvek48e.pdf

A gap sequence with gaps bigger than the hadamard's

/pdf/on-the-central-limit-theorem-for-lacunary-trigonometric-20touy241p.pdf

/pdf/on-the-lacunary-fourier-series-2bxsrx0tdr.pdf

Shigeru Takahashi

Papers

On the law of the iterated logarithm for lacunary trigonometric series, II

The law of the iterated logarithm for a gap sequence with infinite gaps

A gap sequence with gaps bigger than the hadamard's

On the central limit theorem for lacunary trigonometric series

On the lacunary Fourier series