# Search Results

References [1] Berkes , I. 1978 On the central limit theorem for lacunary trigonometric series Anal. Math

## Abstract

We give a new proof of the central limit theorem for one dimensional symmetric random walk in random environment. The proof is quite elementary and natural. We show the convergence of the generators and from this we conclude the convergence of the process. We also investigate the hydrodynamic limit (HDL) of one dimensional symmetric simple exclusion in random environment and prove stochastic convergence of the scaled density field. The macroscopic behaviour of this field is given by a linear heat equation. The diffusion coefficient is the same as that of the corresponding random walk.

Summary This article provides a glimpse of some of the highlights of the joint work of Endre Csáki and Pál Révész since 1979. The topics of this short exploration of the rich stochastic milieu of this inspiring collaboration revolve around Brownian motion, random walks and their long excursions, local times and additive functionals, iterated processes, almost sure local and global central limit theorems, integral functionals of geometric stochastic processes, favourite sites--favourite values and jump sizes for random walk and Brownian motion, random walking in a random scenery, and large void zones and occupation times for coalescing random walks.

BERKES, I., On the almost sure central limit 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

## 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.

## Abstract

Modified least squares processes (MLSP’s) and self-randomized MLSP’s are introduced in *D*[0, 1] for the slope in linear structural and functional error-in-variables models (EIVM’s). Sup-norm approximations in probability
and, as a consequence, functional central limit theorems (CLT’s) are established for the data-based self-normalized versions
of these MLSP’s and self-randomized MLSP’s. The MLSP’s are believed to be new types of objects of study, and the invariance
principles for them constitute new asymptotics, in EIVM’s. Moreover, the obtained data-based functional CLT’s for the MLSP’s
open up new possibilities for constructing various asymptotic confidence intervals (CI’s) for the slope that are named functional
asymptotic CI’s here. Three special examples of such CI’s are given.

## Abstract

The uncitedness factor of a journal is its fraction of uncited articles. Given a set of journals (e.g. in a field) we can determine the rank-order distribution of these uncitedness factors. Hereby we use the Central Limit Theorem which is valid for uncitedness factors since it are fractions, hence averages. A similar result was proved earlier for the impact factors of a set of journals. Here we combine the two rank-order distributions, hereby eliminating the rank, yielding the functional relation between the impact factor and the uncitedness factor. It is proved that the decreasing relation has an S-shape: first convex, then concave and that the inflection point is in the point (μ′, μ) where μ is the average of the impact factors and μ′ is the average of the uncitedness factors.

# Why with bibliometrics the Humanities does not need to be the weakest link

## Indicators for research evaluation based on citations, library holdings, and productivity measures

## Abstract

In this study an attempt is made to establish new bibliometric indicators for the assessment of research in the Humanities. Data from a Dutch Faculty of Humanities was used to provide the investigation a sound empirical basis. For several reasons (particularly related to coverage) the standard citation indicators, developed for the sciences, are unsatisfactory. Target expanded citation analysis and the use of oeuvre (lifetime) citation data, as well as the addition of library holdings and productivity indicators enable a more representative and fair assessment. Given the skew distribution of population data, individual rankings can best be determined based on log transformed data. For group rankings this is less urgent because of the central limit theorem. Lifetime citation data is corrected for professional age by means of exponential regression.

almost sure central limit theorem under minimal conditions, Stat Probab. Letters , 37 (1998), 67–76. Horváth L. An almost sure central limit theorem under minimal conditions