Related papers: Equidistribution, Uniform distribution: a probabil…
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…
A $k$-modal probability distribution over the discrete domain $\{1,...,n\}$ is one whose histogram has at most $k$ "peaks" and "valleys." Such distributions are natural generalizations of monotone ($k=0$) and unimodal ($k=1$) probability…
We elaborate the notions of Martin-L\"of and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform…
Science in the 21st century seems to be governed by novel approaches involving interdisciplinary work, systemic perspectives and complexity theory concepts. These new paradigms force us to leave aside our elder mechanistic approaches and…
\cite{tsagris2025a} proposed the generalized circular projected Cauchy (GCPC) distribution, whose special case is the wrapped Cauchy distribution. In this paper we first derive the relationship with the wrapped Cauchy distribution, and then…
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous…
The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a…
The toss of a coin is usually regarded as the epitome of randomness, and has been used for ages as a means to resolve disputes in a simple, fair way. Perhaps as ancient as consulting objects such as coins and dice is the art of maliciously…
Commonly observed patterns typically follow a few distinct families of probability distributions. Over one hundred years ago, Karl Pearson provided a systematic derivation and classification of the common continuous distributions. His…
The purpose of this article is twofold: to prove a pointwise equidistribution theorem with an error rate for almost smooth functions, which strengthens the main result of Kleinbock, Shi and Weiss (2017); and to obtain a L\'evy-Khintchin…
Counting experiments often rely on Monte Carlo simulations for predictions of Poisson expectations. The accompanying uncertainty from the finite Monte Carlo sample size can be incorporated into parameter estimation by modifying the Poisson…
In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function…
We study a construction published by Donald Knuth in 1965 yielding a completely uniformly distributed sequence of real numbers. Knuth's work is based on de Bruijn sequences of increasing orders and alphabet sizes, which grow exponentially…
Conventional statistics begins with a model, and assigns a likelihood of obtaining any particular set of data. The opposite approach, beginning with the data and assigning a likelihood to any particular model, is explored here for the case…
We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type…
The Generalized Poisson Distribution (GPD) was introduced by Consul and Jain (1973). However, as remarked by Consul (1989), "It is very difficult to prove by direct summation that the sum of all the probabilities is unity". We give a…
Kolmogorov's first axiom of probability is probability takes values between 0 and 1; however, in Cox's derivation of probability having a maximum value of unity is arbitrary since he derives probability as a tool to rank degrees of…
For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this…
Finite-state dimension, introduced early in this century as a finite-state version of classical Hausdorff dimension, is a quantitative measure of the lower asymptotic density of information in an infinite sequence over a finite alphabet, as…
The Weak Law of Large Numbers is traced chronologically from its inception as Jacob Bernoulli's Theorem in 1713, through De Moivre's Theorem, to ultimate forms due to Uspensky and Khinchin in the 1930s, and beyond. Both aspects of Jacob…