Related papers: A Short Tale of Long Tail Integration
In several applications, ultimately at the largest data, truncation effects can be observed when analysing tail characteristics of statistical distributions. In some cases truncation effects are forecasted through physical models such as…
In this paper we present a new family of rules for numerical integration. This family has up to half the error of the widely used Newton-Cotes rules when a sufficient number of points is evaluated and also much better numerical stability…
This paper develops asymptotic approximations of $P(\int_Te^{f(t)}\,dt>b)$ as $b\rightarrow\infty$ for a homogeneous smooth Gaussian random field, $f$, living on a compact $d$-dimensional Jordan measurable set $T$. The integral of an…
This article proposes a new method of truncated estimation to estimate the tail index $\alpha$ of the extremely heavy-tailed distribution with infinite mean or variance. We not only present two truncated estimators $\hat{\alpha}$ and…
Kernel approximation with exponentials is useful in many problems with convolution quadrature and particle interactions such as integral-differential equations, molecular dynamics and machine learning. This paper proposes a weighted…
This article is concerned with a new method for the approximate evaluation of Fourier sine and cosine transforms. We develop and analyse a new quadrature rule for Fourier sine and cosine transforms involving transforming the integral to one…
The paper considers truncation errors for functions of the form $f(x_1,x_2,\dots)=g(\sum_{j=1}^\infty x_j\,\xi_j)$, i.e., errors of approximating $f$ by $f_k(x_1,\dots,x_k)=g(\sum_{j=1}^k x_j\,\xi_j)$, where the numbers $\xi_j$ converge to…
We study inequalities between general integral moduli of continuity of a function and the tail integral of its Fourier transform. We obtain, in particular, a refinement of a result due to D. B. H. Cline [2] (Theorem 1.1 below). We note that…
We offer a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we…
We establish rigorous error bounds for prime counting using a truncated Gaussian (TG) kernel in the explicit formula framework. Our main theorem proves that the approximation error remains globally below 1/2 for all sufficiently large…
Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative…
We study the conformal bootstrap of 1D CFTs on the straight Maldacena-Wilson line in 4D ${\cal N}=4$ super-Yang-Mills theory. We introduce an improved truncation scheme with an 'OPE tail' approximation and use it to reproduce the…
The problem of estimating single- and multi-dimensional integrals, with or without end-point singularities, is prevalent in all fields of scientific research, and in particular in physics. Although tanh-sinh quadrature is known to handle…
The method-of-moments implementation of the electric-field integral equation (EFIE) yields many code-verification challenges due to the various sources of numerical error and their possible interactions. Matters are further complicated by…
In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis.…
Long-time-integrated quantities in stochastic processes, in or out of equilibrium, usually exhibit rare but huge fluctuations. Work or heat production is such a quantity, of which the probability distribution function displays an…
The fine-tuning paradigm in addressing long-tail learning tasks has sparked significant interest since the emergence of foundation models. Nonetheless, how fine-tuning impacts performance in long-tail learning was not explicitly quantified.…
We propose an analytical approach to the computation of tail probabilities of compound distributions whose individual components have heavy tails. Our approach is based on the contour integration method, and gives rise to a representation…
Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for numerical integration on a compact interval. The integrand is assumed to be twice continuously differentiable for the midpoint and trapezoidal rules, and to be…
A highly anticipated use of quantum computers is the simulation of complex quantum systems including molecules and other many-body systems. One promising method involves directly applying a linear combination of unitaries (LCU) to…