[en] Considering a general diffusion process that runs over the non-negative half-line, this paper addresses the first-passage time (FPT) to the origin: the time it takes the process to get from an arbitrary fixed positive level to the level zero. Inspired by the special features of Brownian motion, three types of FPT selfsimilarity are introduced: (i) stochastic, which holds in ‘real space’; (ii) Laplace, which holds in ‘Laplace space’; and (iii) joint, which is the combination of the stochastic and Laplace types. Analysis establishes that the three types of FPT selfsimilarity yield, respectively and universally, the following FPT distributions: inverse-gamma; inverse-Gauss; and Levy–Smirnov. Moreover, the analysis explicitly pinpoints the classes of diffusion processes that produce the three types of selfsimilar FPTs. Shifting from general diffusion dynamics to Langevin dynamics, it is shown that the three classes collapse, respectively, to the following specific processes: diffusion in a logarithmic potential; Brownian motion with drift; and Brownian motion. Also, the effect of the Girsanov transformation on the three types of selfsimilar FPTs is investigated, as well as the effect of initiating the diffusion process from its steady-state level (rather than from a fixed positive level). This paper presents a novel approach to the exploration of first-passage times. (paper)

[en] The exponential, the normal, and the Poisson statistical laws are of major importance due to their universality. Harmonic statistics are as universal as the three aforementioned laws, but yet they fall short in their ‘public relations’ for the following reason: the full scope of harmonic statistics cannot be described in terms of a statistical law. In this paper we describe harmonic statistics, in their full scope, via an object termed harmonic Poisson process: a Poisson process, over the positive half-line, with a harmonic intensity. The paper reviews the harmonic Poisson process, investigates its properties, and presents the connections of this object to an assortment of topics: uniform statistics, scale invariance, random multiplicative perturbations, Pareto and inverse-Pareto statistics, exponential growth and exponential decay, power-law renormalization, convergence and domains of attraction, the Langevin equation, diffusions, Benford’s law, and 1/f noise. - Highlights: • Harmonic statistics are described and reviewed in detail. • Connections to various statistical laws are established. • Connections to perturbation, renormalization and dynamics are established.

[en] Rank distributions are collections of positive sizes ordered either increasingly or decreasingly. Many decreasing rank distributions, formed by the collective collaboration of human actions, follow an inverse power-law relation between ranks and sizes. This remarkable empirical fact is termed Zipf’s law, and one of its quintessential manifestations is the demography of human settlements — which exhibits a harmonic relation between ranks and sizes. In this paper we present a comprehensive statistical-physics analysis of rank distributions, establish that power-law and exponential rank distributions stand out as optimal in various entropy-based senses, and unveil the special role of the harmonic relation between ranks and sizes. Our results extend the contemporary entropy-maximization view of Zipf’s law to a broader, panoramic, Gibbsian perspective of increasing and decreasing power-law and exponential rank distributions — of which Zipf’s law is one out of four pillars

[en] Markov dynamics constitute one of the most fundamental models of random motion between the states of a system of interest. Markov dynamics have diverse applications in many fields of science and engineering, and are particularly applicable in the context of random motion in networks. In this paper we present a two-dimensional gauging method of the randomness of Markov dynamics. The method–termed Markov Stochasticity Coordinates–is established, discussed, and exemplified. Also, the method is tweaked to quantify the stochasticity of the first-passage-times of Markov dynamics, and the socioeconomic equality and mobility in human societies.

[en] In this paper we introduce and analyze the Poisson Aggregation Process (PAP): a stochastic model in which a random collection of random balls is stacked over a general metric space. The scattering of the balls’ centers follows a general Poisson process over the metric space, and the balls’ radii are independent and identically distributed random variables governed by a general distribution. For each point of the metric space, the PAP counts the number of balls that are stacked over it. The PAP model is a highly versatile spatial counterpart of the temporal M/G/∞ model in queueing theory. The surface of the moon, scarred by circular meteor-impact craters, exemplifies the PAP model in two dimensions: the PAP counts the number of meteor-impacts that any given moon-surface point sustained. A comprehensive analysis of the PAP is presented, and the closed-form results established include: general statistics, stationary statistics, short-range and long-range dependencies, a Central Limit Theorem, an Extreme Limit Theorem, and fractality.

[en] Over 80 years ago Samuel Wilks proposed that the “generalized variance” of a random vector is the determinant of its covariance matrix. To date, the notion and use of the generalized variance is confined only to very specific niches in statistics. In this paper we establish that the “Wilks standard deviation” –the square root of the generalized variance–is indeed the standard deviation of a random vector. We further establish that the “uncorrelation index” –a derivative of the Wilks standard deviation–is a measure of the overall correlation between the components of a random vector. Both the Wilks standard deviation and the uncorrelation index are, respectively, special cases of two general notions that we introduce: “randomness measures” and “independence indices” of random vectors. In turn, these general notions give rise to “randomness diagrams”—tangible planar visualizations that answer the question: How random is a random vector? The notion of “independence indices” yields a novel measure of correlation for Lévy laws. In general, the concepts and results presented in this paper are applicable to any field of science and engineering with random-vectors empirical data.

[en] Lévy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts of the Generalized CLT, thus establishing a geometric theory for Lévy distributions. The theory explains the universal emergence of Lévy distributions in physical settings which are well beyond the realm of the Generalized CLT

[en] This paper addresses symmetric Ito diffusions, with general position-dependent velocities and diffusion coefficients, that run over the real line. The paper establishes an explicit characterization of diffusions that are selfsimilar processes with arbitrary Hurst exponents. Specifically, for a given diffusion, the paper asserts that the four following statements are equivalent: (i) the diffusion’s trajectories are selfsimilar; (ii) the diffusion’s positions display a certain scaling property; (iii) the diffusion’s first-passage times (FTPs) to the origin display a certain scaling property; (iv) the diffusion’s velocities and diffusion coefficients admit certain power-law forms. For diffusions with constant diffusion coefficients, the paper further asserts that: selfsimilarity holds if and only if the underpinning potential is logarithmic, in which case the Hurst exponent is $\frac{1}{2}$. The positions of selfsimilar diffusions are shown to have non-Gaussian statistics with the following features: densities that are either unimodal and explosive, or unimodal, or bimodal; and tails that are either ‘light’, or Gamma, or ‘heavy’. The FTPs to the origin of selfsimilar diffusions are shown to have inverse-Gamma statistics with infinite means and infinite moments. The results presented in this paper unveil a class of Markovian diffusions, whose trajectories are fractal objects, and whose non-Gaussian statistics can be custom designed via three parameters: a velocity parameter, a diffusion parameter, and a Hurst exponent. (letter)

[en] Random vectors with a symmetric correlation structure share a common value of pair-wise correlation between their different components. The symmetric correlation structure appears in a multitude of settings, e.g. mixture models. In a mixture model the components of the random vector are drawn independently from a general probability distribution that is determined by an underlying parameter, and the parameter itself is randomized. In this paper we study the overall correlation of high-dimensional random vectors with a symmetric correlation structure. Considering such a random vector, and terming its pair-wise correlation “micro-correlation”, we use an asymptotic analysis to derive the random vector’s “macro-correlation” : a score that takes values in the unit interval, and that quantifies the random vector’s overall correlation. The method of obtaining macro-correlations from micro-correlations is then applied to a diverse collection of frameworks that demonstrate the method’s wide applicability.

[en] As noise is omnipresent, real-world quantities measured by scientists and engineers are commonly obtained in the form of statistical distributions. In turn, perhaps the most compact representation of a given statistical distribution is via the mean-variance approach: the mean manifesting the distribution’s ‘typical’ value, and the variance manifesting the magnitude of the distribution’s fluctuations about its mean. The mean-variance approach is based on an underlying Euclidean-geometry perspective. So very often real-world quantities of interest are non-negative sizes, and their measurements yield statistical size distributions. In this paper, and in the context of size distributions, we present an alternative to the Euclidean-based mean-variance approach: a mean-equality approach that is based on an underlying socioeconomic perspective. We establish two equality indices that score, on a unit-interval scale, the intrinsic ‘egalitarianism’ of size distributions: (i) the poverty equality index which is particularly sensitive to the existence of very small “poor” sizes; (ii) the riches equality index which is particularly sensitive to the existence of very large “rich” sizes. These equality indices, their properties, their computation, their application, and their connections to the mean-variance approach – are explored and described comprehensively.