Related papers: Quantitative uniform distribution results for geom…

Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic…

The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to [2,\infty)$, we determine…

The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note, we use the method of moments to study the distribution of…

We give an extension of a criterion of van der Corput on uniform distribution of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions $\Delta^2x_n\to…

In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed sequences of partitions. Among them we found a large class having low discrepancy (which means of order 1/N. One of them is the…

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special…

Let $X_1, \ldots, X_n$ be independent non-negative random variables with cumulative distribution functions $F_1,F_2,\ldots,F_n$, each satisfying certain (rather mild) conditions. We show that the median of $k$-th smallest order statistic of…

The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $(\lambda_t)_{t \in [0,1]}$ of invariant extensions of the linear Lebesgue measure $\lambda$ in…

The classical theorem of Birkhoff states that the $T^N f(x) = (1/N)\sum_{k=0}^{N-1} f(\sigma^k x)$ converges almost everywhere for $x\in X$ and $f\in L^{1}(X)$, where $\sigma$ is a measure preserving transformation of a probability measure…

Gaussian distributions can be generalized from Euclidean space to a wide class of Riemannian manifolds. Gaussian distributions on manifolds are harder to make use of in applications since the normalisation factors, which we will refer to as…

Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given…

Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$…

Let $M_n^{(k)}$ denote the $k$th largest maximum of a sample $(X_1,X_2,...,X_n)$ from parent $X$ with continuous distribution. Assume there exist normalizing constants $a_n>0$, $b_n\in \mathbb{R}$ and a nondegenerate distribution $G$ such…

We show that if an essentially arbitrary sequence supported on an interval containing $x$ integers, is convolved with a tiny Siegel-Walfisz-type sequence supported on an interval containing $\exp((\log x)^{\varepsilon})$ integers then the…

This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first part is concerned with sequences of the type x_k = n_k*alpha, n_1 < n_2 < n_3 < ..., mod 1. Improving a result of Salat we…

Let $X_1,\ldots,X_n$ be an i.i.d. sample from symmetric stable distribution with stability parameter $\alpha$ and scale parameter $\gamma$. Let $\varphi_n$ be the empirical characteristic function. We prove an uniform large deviation…

Let $K(=K_{n,\theta})$ be a positive integer-valued random variable whose distribution is given by ${\rm P}(K = x) = \bar{s}(n,x) \theta^x/(\theta)_n$ $(x=1,\ldots,n) $, where $\theta$ is a positive number, $n$ is a positive integer,…

This paper develops unified asymptotic distribution theory for dynamic quantile predictive regressions which is useful when examining quantile predictability in stock returns under possible presence of nonstationarity.

In 1931, Van der Corput showed that if for each positive integer $s$, the sequence $\{x_{n+s}-x_n\}$ is uniformly distributed (mod 1), then the sequence $x_n$ is uniformly distributed (mod 1). The converse of above result is surprisingly…

In this paper, we give a general formula to determine the quantization coefficients for uniform distributions defined on the boundaries of different regular $m$-sided polygons inscribed in a circle. The result shows that the quantization…