Related papers: On the long tail property of product convolution
A record is an entry in a time series that is larger or smaller than all previous entries. If the time series consists of independent, identically distributed random variables with a superimposed linear trend, record events are positively…
We introduce a general class of continuous univariate distributions with positive support obtained by transforming the class of two-piece distributions. We show that this class of distributions is very flexible, easy to implement, and…
Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the…
There are two kinds of Tsallis-probability distributions: heavy tail ones and compact support distributions. We show here, appealing to functional analysis' tools, that for lower bound Hamiltonians only the first type guarantees a maximum…
We study the empirical version of halfspace depths with the objective of establishing a connection between the rates of convergence and the tail behaviour of the corresponding underlying distributions. The intricate interplay between the…
In this work, a general definition of Convolution between two arbitrary Tempered Ultradistributions is given. When one of the Tempered Ultradistributions is rapidly decreasing this definition coincides with the definition of J. Sebastiao e…
Levy flights in the space of mutations model time evolution of bacterial DNA. Parameters in the model are adjusted in order to fit observations coming from the Long Time Evolution Experiment with E. Coli.
We discuss non-Gaussian random matrices whose elements are random variables with heavy-tailed probability distributions. In probability theory heavy tails of the distributions describe rare but violent events which usually have dominant…
Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of…
We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the…
We introduce a large and flexible class of discrete tempered stable distributions, and analyze the domains of attraction for both this class and the related class of positive tempered stable distributions. Our results suggest that these are…
The long-tail phenomenon tells us that there are many items in the tail. However, not all tail items are the same. Each item acquires different kinds of users. Some items are loved by the general public, while some items are consumed by…
Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the product $XY$ is derived. Some basic distributional properties are also derived, including…
Condensation is the phenomenon whereby one of a sum of random variables contributes a finite fraction to the sum. It is manifested as an aggregation phenomenon in diverse physical systems such as coalescence in granular media, jamming in…
Let $S_n$ be the sum of independent random variables with distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$ \lim_{n\to\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1\right| =0 $$ are…
We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In…
Regular variation is often used as the starting point for modeling multivariate heavy-tailed data. A random vector is regularly varying if and only if its radial part $R$ is regularly varying and is asymptotically independent of the angular…
How to estimate the uncertainty of a given model is a crucial problem. Current calibration techniques treat different classes equally and thus implicitly assume that the distribution of training data is balanced, but ignore the fact that…
Recent theoretical studies have shown that heavy-tails can emerge in stochastic optimization due to `multiplicative noise', even under surprisingly simple settings, such as linear regression with Gaussian data. While these studies have…
This note presents an operational measure of fat-tailedness for univariate probability distributions, in $[0,1]$ where 0 is maximally thin-tailed (Gaussian) and 1 is maximally fat-tailed. Among others,1) it helps assess the sample size…