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Since its introduction by J. Karamata, regular variation has evolved from a purely mathematical concept into a cornerstone of theoretical probability and data analysis. It is extensively studied and applied in different areas. Its…
Starting from the BBGKY hierarchy, describing the kinetics of nonlinear particle system, we obtain the relevant entropy and stationary distribution function. Subsequently, by employing the Lorentz transformations we propose the relativistic…
We study the spectrum of primordial fluctuations in theories where the inflaton field is coupled to massless fields and/or to itself. Conformally invariant theories generically predict a scale invariant spectrum. Scales entering the theory…
An empirical model is a generalization of a probability space. It consists of a simplicial complex of subsets of a class X of random variables such that each simplex has an associated probability distribution. The ensuing marginalizations…
Using the concept of constant evasion to different sorts of suitable binary relations, we establish many cardinal invariants derived from the established cardinal invariants $\mathfrak{e}^\mathrm{const}_{n}$ and…
Symmetries play a conspicuous role in the large-scale behavior of critical systems. While in equilibrium they allow to classify asymptotics into different universality classes, out of equilibrium they can emerge, some times unexpectedly, as…
When inferring parameters from a Gaussian-distributed data set by computing a likelihood, a covariance matrix is needed that describes the data errors and their correlations. If the covariance matrix is not known a priori, it may be…
Taylors Law (TL) describes the scaling relationship between the mean and variance of populations as a power-law. TL is widely observed in ecological systems across space and time with exponents varying largely between 1 and 2. Many…
This Colloquium reviews statistical models for money, wealth, and income distributions developed in the econophysics literature since the late 1990s. By analogy with the Boltzmann-Gibbs distribution of energy in physics, it is shown that…
Quantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are…
We consider a multinomial distribution, where the number of cells increases and the cell-probabilities decreases as the number of observations grows. The probabilities of large deviations of statistics, which has form of a sum of Borel…
Permutations of correlated sequences of random variables appear naturally in a variety of applications such as graph matching and asynchronous communications. In this paper, the asymptotic statistical behavior of such permuted sequences is…
The frequency of occurrence of prime numbers at unit number spacing intervals exhibits selfsimilar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows,…
We look into the fluctuations caused by disturbances in power systems. In the linearized system of the power systems, the disturbance is modeled by a Brownian motion process, and the fluctuations are described by the covariance matrix of…
We report on the computer study of a lattice system that relaxes from a metastable state. Under appropriate nonequilibrium randomness, relaxation occurs by avalanches, i.e., the model evolution is discontinuous and displays many scales in a…
We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of $\sqrt n$ and directional statistics for a shifted lattice. We show that the randomly rotated, and…
We propose a probability distribution for multivariate binary random variables. The probability distribution is expressed as principal minors of the parameter matrix, which is a matrix analogous to the inverse covariance matrix in the…
The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is esablished. A set of combinations of expectation values whose value does not in general depend…
The fluctuation scaling law has universally been observed in a wide variety of phenomena. For counting processes describing the number of events occurred during time intervals, it is expressed as a power function relationship between the…
We develop stochastic variational inference, a scalable algorithm for approximating posterior distributions. We develop this technique for a large class of probabilistic models and we demonstrate it with two probabilistic topic models,…