Related papers: Local Dvoretzky-Kiefer-Wolfowitz confidence bands

Estimates of finite population cumulativedistribution functions (CDFs) and quantiles are critical forpolicy-making, resource allocation, and public health planning. For instance, federal finance agencies may require accurate estimates of…

Let $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$, and $F$ the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$. We show that stationarity of $\{X_n\}_{n\geq 1}$…

A quantile is defined as a value below which random draws from a given distribution falls with a given probability. In a centralized setting where the cumulative distribution function (CDF) is unknown, the empirical CDF (ECDF) can be used…

An index of uniformity is developed as an alternative to the maximum-entropy principle for selecting continuous, differentiable probability distributions $\mathcal{P}$ subject to constraints $C$. The uniformity index developed in this paper…

This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional…

Let $d$ be a probability distribution. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{d^{*n}(x)}{n}=1,\qquad\text{where}\quad d^{*n}:=\underbrace{\,d*d*\cdots*d\,}_{n\text{ times}}. $$ For a…

A notion of local $U$-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for $U$-processes that may be useful in other contexts. This local…

We establish a universal framework for concentration inequalities based on invariance under diffeomorphism groups. Given a probability measure $\mu$ on a space $E$ and a diffeomorphism $\psi: E \to F$, concentration properties transfer…

The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the $d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions on the distribution, the…

Let $X_1,\ldots,X_n$ be an i.i.d. sample from symmetric stable distribution with stability parameter $\alpha$ and scale parameter $\gamma$. Let $\varphi_n$ be the empirical characteristic function. We prove an uniform large deviation…

We consider a (sub)diffusion equation with a nonlinearity of the form $pf(u)-qu$, where $p$ and $q$ are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-Kamenetskii-Zeldovich and the Allen-Cahn equations. We…

The use of summary statistics beyond the two-point correlation function to analyze the non-Gaussian clustering on small scales is an active field of research in cosmology. In this paper, we explore a set of new summary statistics -- the…

We consider vanishing properties of exponential sums of the Liouville function $\lambda$ of the form $$ \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq…

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the…

Let $\mathcal{F}$ be a class of measurable functions $f:S\mapsto [0,1]$ defined on a probability space $(S,\mathcal{A},P)$. Given a sample (X_1,...,X_n) of i.i.d. random variables taking values in S with common distribution P, let P_n…

We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this…

In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian--Caputo fractional…

We provide sufficient conditions under which the center-outward distribution and quantile functions introduced in Chernozhukov et al.~(2017) and Hallin~(2017) are homeomorphisms, thereby extending a recent result by Figalli \cite{Fi2}. Our…

We show that under minimal assumptions on a class of functions $\mathcal{H}$ defined on a probability space $(\mathcal{X},\mu)$, there is a threshold $\Delta_0$ satisfying the following: for every $\Delta\geq\Delta_0$, with probability at…

If $\mu$ is a distribution over the $d$-dimensional Boolean cube $\{0,1\}^d$, our goal is to estimate its mean $p\in[0,1]^d$ based on $n$ iid draws from $\mu$. Specifically, we consider the empirical mean estimator $\hat p_n$ and study the…