Related papers: Spacing Distribution of a Bernoulli Sampled Sequen…
We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum $S_n=\sum_{j=1}^nX_{j,\pi(j)}$ of a random $n\times n$ matrix $X=(X_{j,r})$, where the $X_{j,r}$ are independent integer valued random…
Let $G$ be a finite connected graph and let $G^{[\star N,k]}$ be the distance $k$-graph of the $N$-fold star power of $G$. For a fixed $k\geq1$, we show that the large $N$ limit of the spectral distribution of $G^{[\star N,k]}$ converges to…
This paper studies the sample complexity of searching over multiple populations. We consider a large number of populations, each corresponding to either distribution P0 or P1. The goal of the search problem studied here is to find one…
We introduce a sharpness functional for probabilistic models that quantifies sharpness as an intrinsic property of the probability distribution. The measure is derived based on a rank-based concentration principle that tracks upward…
The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting,…
One can consider $\mu$-Martin-L\"of randomness for a probability measure $\mu$ on $2^{\omega}$, such as the Bernoulli measure $\mu_p$ given $p \in (0, 1)$. We study Bernoulli randomness of sequences in $n^{\omega}$ with parameters $p_0,…
Let n points be taken at random on a circle of unit circumference and clockwise ordered. Uniform spacings are defined as the clockwise arc-lengths between the successive points from this sample. We are interested in the asymptotic behavior…
We survey recent results beyond equidistribution of sequences modulo one. We focus on the sequence of angles in a Euclidean lattice in $\mathbb R^2$ and on the sequence $\sqrt n\bmod1 $.
Given a real number beta > 1, the spectrum of beta is a well studied dynamical object. In this article we show the existence of a certain measure on the spectrum of beta related to the distribution of random polynomials in beta, and discuss…
The family of skew-symmetric distributions is a wide set of probability density functions obtained by combining in a suitable form a few components which are selectable quite freely provided some simple requirements are satisfied. Intense…
The distribution of the consecutive level-spacing ratio is now widely used as a tool to distinguish integrable from chaotic quantum spectra, mostly due to its avoiding of the numerical spectral unfolding. Similar to the use of the…
Let $A^{(n)}_{l;k}\subset S_n$ denote the set of permutations of $[n]$ for which the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions. Under the uniformly probability measure $P_n$ on…
We study different ways of determining the mean distance $ < r_n >$ between a reference point and its $n$-th neighbour among random points distributed with uniform density in a $D$-dimensional Euclidean space. First we present a heuristic…
The negative binomial distribution NB$(k,r)$ of Type I is the probability distribution for a sequence of independent Bernoulli trials (with success parameter $p\in(0,1)$) with $r$ nonoverlapping success runs of length $\ge k$. We present a…
We investigate spacing statistics $p(s)$ and distribution of eigenvalues $D(\epsilon)$ for ensembles of various real random matrices (of order $n \times n, n=2$ and $n>>2$) where the matrix-elements have various Probability Distribution…
For a given one-dimensional random walk $\{S_n\}$ with a subexponential step-size distribution, we present a unifying theory to study the sequences $\{x_n\}$ for which $\mathsf{P}\{S_n>x\}\sim n\mathsf{P}\{S_1>x\}$ as $n\to\infty$ uniformly…
Consider an $N\times N$ hermitian random matrix with independent entries, not necessarily Gaussian, a so called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between…
We study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space. We let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this…
Sequencing by synthesis is used in many next-generation DNA sequencing technologies. Some of the technologies, especially those exploring the principle of single-molecule sequencing, allow incomplete nucleotide incorporation in each cycle.…
We study the asymptotic behavior of the sums of divisors when the integers are modelled with the Bernoulli random walk; We prealably study the correlation properties of the corresponding system.