Related papers: Testing statistical laws in complex systems
A new version of a strong law of large numbers for a ``good'' pairwise independent sequence of random variables (r.v.'s) with a small part of ``bad'' dependent r.v.'s is proposed. The main goal is to relax the assumption on the existence of…
Zipf's law of abbreviation, the tendency of more frequent words to be shorter, is one of the most solid candidates for a linguistic universal, in the sense that it has the potential for being exceptionless or with a number of exceptions…
Zipf's power-law distribution is a generic empirical statistical regularity found in many complex systems. However, rather than universality with a single power-law exponent (equal to 1 for Zipf's law), there are many reported deviations…
Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all…
Power-law probability distributions arise often in the social and natural sciences. Statistics have been developed for estimating the exponent parameter as well as gauging goodness-of-fit to a power law. Yet paradoxically, many famous power…
The spatial distribution of people exhibits clustering across a wide range of scales, from household ($\sim 10^{-2}$ km) to continental ($\sim 10^4$ km) scales. Empirical data indicates simple power-law scalings for the size distribution of…
In response to growing concern about the reliability and reproducibility of published science, researchers have proposed adopting measures of greater statistical stringency, including suggestions to require larger sample sizes and to lower…
Avalanche statistics of various threshold activated dynamical systems are known to depend on the magnitude of the drive, or stress, on the system. Such dependencies exist for earthquake size distributions, in sheared granular avalanches,…
Consider $n$ independent measurements, with the additional information of the times at which measurements are performed. This paper deals with testing statistical hypotheses when $n$ is large and only a small amount of observations…
The Strong Law of Large Numbers (SLLN) for random variables or random vectors with different mathematical expectations easily reduces by means of shifts to SLLN for random variables or random vectors whose mathematical expectations are…
This paper investigates a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors, that is,…
Prediction in complex systems at criticality is believed to be very difficult, if not impossible. Of particular interest is whether earthquakes, whose distribution follows a power law (Gutenberg-Richter) distribution, are in principle…
We apply recent ideas about complexity and randomness to the philosophy of laws and chances. We develop two ways to use algorithmic randomness to characterize probabilistic laws of nature. The first, a generative chance* law, employs a…
In this paper we obtain some possibilistic variants of the probabilistic laws of large numbers, different from those obtained by other authors, but very natural extensions of the corresponding ones in probability theory. Our results are…
We study the law of the iterated logarithm (LIL) for the maximum likelihood estimation of the parameters (as a convex optimization problem) in the generalized linear models with independent or weakly dependent ($\rho$-mixing, $m$-dependent)…
It has been shown recently that a specific class of path-dependent stochastic processes, which reduce their sample space as they unfold, lead to exact scaling laws in frequency and rank distributions. Such Sample Space Reducing processes…
When following a sequence - such as reading a text or tracking a user's activity - one can measure how the "dictionary" of distinct elements (types) grows with the number of observations (tokens). When this growth follows a power law, it is…
In order to clarify how the statistical properties of earthquakes depend on the constitutive law characterizing the stick-slip dynamics, we make an extensive numerical simulation of the one-dimensional spring-block model with the rate- and…
As the extension of uncorrelated single-valued random variables, set-valued case is studied in this paper. When the underlying space is of finite dimension, by using the support function, We shall prove the weak and strong laws of large…
Statistical physics is employed to evaluate the performance of error-correcting codes in the case of finite message length for an ensemble of Gallager's error correcting codes. We follow Gallager's approach of upper-bounding the average…