Related papers: Relating Turing's Formula and Zipf's Law
We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy…
We prove asymptotic formulas for the complex coefficients of $(\zeta q;q)_\infty^{-1}$, where $\zeta$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions…
In this work, we attempt to refine the classic asymptotic formulae to describe the probability distribution of likelihood-ratio statistical tests. The idea is to split the probability distribution function into two parts. One part is…
The article introduces corrections to Zipf's and Heaps' laws based on systematic models of the proportion of hapaxes, i.e., words that occur once. The derivation rests on two assumptions: The first one is the standard urn model which…
Clarke and Barron have recently shown that the Jeffreys' invariant prior of Bayesian theory yields the common asymptotic (minimax and maximin) redundancy of universal data compression in a parametric setting. We seek a possible analogue of…
Motivated by applications to group synchronization and quadratic assignment on random data, we study a general problem of Bayesian inference of an unknown ``signal'' belonging to a high-dimensional compact group, given noisy pairwise…
Background: Zipf's law and Heaps' law are observed in disparate complex systems. Of particular interests, these two laws often appear together. Many theoretical models and analyses are performed to understand their co-occurrence in real…
The system of equations for parametric sub-resonant growth of the amplitude of oscillations was obtained. The time of turning point from the growing of the amplitude to the bounded oscillations in the slow variable was found. The comparison…
This paper studies the limits of language models' statistical learning in the context of Zipf's law. First, we demonstrate that Zipf-law token distribution emerges irrespective of the chosen tokenization. Second, we show that Zipf…
We model the generation of words with independent unequal probabilities of occurrence of letters. We prove that the probability $p(r)$ of occurrence of words of rank $r$ has a power asymptotics. As distinct from the paper published earlier…
The distribution of the population of cities has attracted a great deal of attention, in part because it sharply constrains models of local growth. However, to this day, there is no consensus on the distribution below the very upper tail,…
Zipf's law is shown to arise as the variational solution of a problem formulated in Fisher's terms. An appropriate minimization process involving Fisher information and scale-invariance yields this universal rank distribution. As an example…
The aim of this paper is to provide a resampling technique that allows us to make inference on superpopulation parameters in finite population setting. Under complex sampling designs, it is often difficult to obtain explicit results about…
The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family…
The prevailing maximum likelihood estimators for inferring power law models from rank-frequency data are biased. The source of this bias is an inappropriate likelihood function. The correct likelihood function is derived and shown to be…
We show that statistical criticality, i.e. the occurrence of power law frequency distributions, arises in samples that are maximally informative about the underlying generating process. In order to reach this conclusion, we first identify…
We demonstrate that large texts, representing human (English, Russian, Ukrainian) and artificial (C++, Java) languages, display quantitative patterns characterized by the Benford-like and Zipf laws. The frequency of a word following the…
We consider the following variant of the classic collector's problem: The family of coupon probabilities is the mixing of two subfamilies one of which is the \textit{uniform} family, while the other belongs to the well known \textit{Zipf…
In this letter we study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit…
Asymptotic expansions are derived for the tail distribution of the product of two correlated normal random variables with non-zero means and arbitrary variances, and more generally the sum of independent copies of such random variables.…