# Search Results

References [1] Berkes , I. , A central limit theorem for trigonometric series with small gaps, Z

theorem and domains of attraction, Probab. Theory Related Fields 102 (1995), 1-17. MR 96j :60033 On the almost sure central limit theorem and domains of attraction Probab. Theory Related Fields

## Summary

We provide uniform rates of convergence in the central limit theorem for linear negative quadrant dependent (LNQD) random
variables. Let

-Sklodowska, Lublin LVI 1 18 Fazekas, I. and Rychlik, Z., Almost sure central limit theorems for random fields, Math

## Abstract

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let *α* be an arbitrary real root of a quadratic equation with integer coefficients; say,
. Given any rational number 0 < *x* < 1 (say, *x* = 1/2) and any positive integer *n*, we count the number of elements of the sequence *α*, 2*α*, 3*α*, ..., *nα* modulo 1 that fall into the subinterval [0, *x*]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected
number” *nx* from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ *n* ≤ *N*. Depending on *α* and *x*, we may need an extra additive correction of constant times logarithm of *N*; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm
of *N*. If *N* is large, the distribution of this renormalized counting number, as n runs in 1 ≤ *n* ≤ *N*, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as
*N* tends to infinity. This is the main result of the paper (see Theorem 1.1).

## Abstract

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let *α* be an arbitrary real root of a quadratic equation with integer coefficients; say, *α* =

*x*< 1 (say,

*x*= 1/2) and any positive integer

*n*, we count the number of elements of the sequence

*α*, 2

*α*, 3

*α*, …,

*nα*modulo 1 that fall into the subinterval [0,

*x*]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number”

*nx*from the counting number, and study the typical fluctuation of this difference as

*n*runs in a long interval 1 ≤

*n*≤

*N*. Depending on

*α*and

*x*, we may need an extra additive correction of constant times logarithm of

*N*; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of

*N*. If

*N*is large, the distribution of this renormalized counting number, as

*n*runs in 1 ≤

*n*≤

*N*, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as

*N*tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on

*α*and

*x*, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums. The whole paper consists of an introduction and 17 sections. Part 1 contains the Introduction and Sections 1–7.

In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.