Related papers: Limit Theorems for Fr\'echet Mean Sets

Two central limit theorems for sample Fr\'echet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first one, Theorem 2.2, encompasses and improves upon most earlier CLTs on Fr\'echet means and…

We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an H\"older potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of…

This paper develops new limit theory for data that are generated by networks or more generally display cross-sectional dependence structures that are governed by observable and unobservable characteristics. Strategic network formation…

We consider the hard-edge scaling of the Mittag-Leffler ensemble confined to a fixed disk inside the droplet. Our primary emphasis is on fluctuations of rotationally-invariant additive statistics that depend on the radius and thus give rise…

The Fr\'echet mean is an important statistical summary and measure of centrality of data; it has been defined and studied for persistent homology captured by persistence diagrams. However, the complicated geometry of the space of…

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical…

This paper develops limit theorems for random variables with network dependence, without requiring the individuals in the network to be located in a Euclidean or metric space. This distinguishes our approach from most existing limit…

We show that if $(X, \mu, T)$ is a probability measure-preserving dynamical system, and $\mathscr{P}$ is a countable partition of $(X, \mu)$, then the limit $$ \lim_{n, k \to \infty} \mathbb{E} \left[ \frac{1}{k} \sum_{j = 0}^{k - 1} f…

Recently a new type of central limit theorem for belief functions was given in Epstein et al. [9]. In this paper, we generalize the central limit theorem in Epstein et al. [9] to accommodate general bounded random variables. These results…

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fr\'{e}chet mean (barycenter) is…

The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical…

We introduce the notion of a random mean generated by a random variable and give a construction of its expected value. We derive some sufficient conditions under which strong laws of large numbers and some limit theorems hold for random…

Under the sublinear expectation $\mathbb{E}[\cdot]:=\sup_{\theta\in \Theta} E_\theta[\cdot]$ for a given set of linear expectations $\{E_\theta: \theta\in \Theta\}$, we establish a new law of large numbers and a new central limit theorem…

The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of…

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L^2-L^p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential…

We prove that for q>=1, there exists r(q)<1 such that for p>r(q), the number of points in large boxes which belongs to the infinite cluster has a normal central limit behaviour under the random cluster measure phi_{p,q} on Z^d, d>=2.…

Central limit theorems (CLTs) have a long history in probability and statistics. They play a fundamental role in constructing valid statistical inference procedures. Over the last century, various techniques have been developed in…

The Glivenko--Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges to the underlying distribution (in the sense of uniform…

Let $(\S^1,d_{\S^1})$ be the unit circle in $\R^2$ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure $\mu$ to admit a well defined Fr\'echet mean on $(\S^1,d_{\S^1})$. %This…

In the spirit of a classical results for Crump-Mode-Jagers processes, we prove a strong law of large numbers for homogenous fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we…