Related papers: Estimation in a fluctuating medium and power-law d…
The statistics of natural catastrophes contains very counter-intuitive results. Using earthquakes as a working example, we show that the energy radiated by such events follows a power-law or Pareto distribution. This means, in theory, that…
The power law is ubiquitous in natural and social phenomena, and is considered as a universal relationship between the frequency and its rank for diverse social systems. However, a general model is still lacking to interpret why these…
We give simple proofs, under minimal hypotheses, of the Weak Law of Large Numbers and the Central Limit Theorem for independent identically distributed random variables. These proofs use only the elementary calculus, together with the most…
This note points out how confidence interval estimates for standard deviation transform into confidence interval estimates for the power of F-tests at fixed alternative means. An application is shown for the test of a two-sided hypothesis…
A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies…
In this paper we explain the wild fluctuations of financial prices from the intrinsic amplifying feedback of speculative supply and demand. Formally, we show that an asset return follows a multiplicative random growth with exogenous input,…
Recently (arXiv:0910.2870), we have derived a fluctuation theorem for systems in thermodynamic equilibrium compatible with anomalous response functions, e.g. the existence of states with \textit{negative heat capacities} $C<0$. In this…
We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We…
Several probability distributions have been proposed in the literature, especially with the aim of obtaining models that are more flexible relative to the behaviors of the density and hazard rate functions. Recently, a new generalization of…
We consider a version of D. Price's model for the growth of a bibliographic network, where in each iteration a constant number of citations is randomly allocated according to a weighted combination of accidental (uniformly distributed) and…
This paper investigates the combinatorics that gives rise to the Boltzmann probability distribution. Despite being one of the most important distributions in physics and other fields of science, the mathematics of the underlying model of…
Determining the power distribution of the members of a shareholder meeting or a legislative committee is a well-known problem for many applications. In some cases it turns out that power is nearly proportional to relative voting weights,…
Over the last decades, impressive progresses have been made in many experimental domains, e.g. microscopic techniques such as single-particle tracking, leading to plethoric amounts of data. In a large variety of systems, from natural to…
We derive a functional central limit theorem (fclt) for normalised sums of a function of the partial sums of independent and identically distributed random variables. In particular, we show, using a technique presented in Huang and Zhang…
We argue that an ensemble of backgrounds best understands hydrodynamic dispersion relations in a medium with few degrees of freedom and is therefore subject to strong thermal fluctuations. In the linearized regime, dispersion relations…
We introduce the notion of a Young generating function for a probability measure on integer partitions. We use this object to characterize probability distributions over integer partitions satisfying a law of large numbers and those that…
We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved…
We determine the cumulants of electromagnetic energy transmitted through one-dimensional disordered medium with absorption or amplification. For this purpose we derive the Keldysh nonlinear sigma-model action with a source term that…
Fluctuation dynamics of an experimentally measured observable offer a primary signal for nonequilibrium systems, along with dynamics of the mean. While universal speed limits for the mean have actively been studied recently, constraints for…
In this article, we proposed a new probability distribution named as power Maxwell distribution (PMaD). It is another extension of Maxwell distribution (MaD) which would lead more flexibility to analyze the data with non-monotone failure…