For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become necessary) for the convergence, in transportation, of probability measures when the cost function is continuous, non-decreasing and depends on the distance. As an application, the CLT in the transportation distance is proved for independent and some dependent stationary sequences.
We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on R d , with d $\ge$ 1. We provide new results on the uniqueness and stability of the associated optimal transportation potentials , namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.
IMUVA, Universidad de Valladolid（巴利亚多利德大学IMUVA数学研究所）；IMT, Université de Toulouse（图卢兹大学图卢兹数学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）
We prove a central limit theorem for the entropic transportation cost between subgaussian probability measures, centered at the population cost. This is the first result which allows for asymptotically valid inference for entropic optimal transport between measures which are not necessarily discrete. In the compactly supported case, we complement these results with new, faster, convergence rates for the expected entropic transportation cost between empirical measures. Our proof is based on strengthening convergence results for dual solutions to the entropic optimal transport problem.
We consider the problem of optimal transportation with general cost between a empirical measure and a general target probability on R d , with d $\ge$ 1. We extend results in [19] and prove asymptotic stability of both optimal transport maps and potentials for a large class of costs in R d. We derive a central limit theorem (CLT) towards a Gaussian distribution for the empirical transportation cost under minimal assumptions, with a new proof based on the Efron-Stein inequality and on the sequential compactness of the closed unit ball in L 2 (P) for the weak topology. We provide also CLTs for e…
We provide a Central Limit Theorem for the Monge-Kantorovich distance between two empirical distributions with size $n$ and $m$, $W_p(P_n,Q_m)$ for $p>1$ for observations on the real line, using a minimal amount of assumptions. We provide an estimate of the asymptotic variance which enables to build a two sample test to assess the similarity between two distributions. This test is then used to provide a new criterion to assess the notion of fairness of a classification algorithm.
We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author, for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) We derive the first non-asymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors; (c) We give an improved bound for…
The notion of entropy-regularized optimal transport, also known as Sinkhorn divergence, has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data analysis. The Sinkhorn divergence allows the fast computation of an entropically regularized Wasserstein distance between two probability distributions supported on a finite metric space of (possibly) high-dimension. For data sampled from one or two unknown probability distributions, we derive the distributional limits of the empirical Sinkhorn divergence and …
In this paper, we give estimates of the quadratic transportation cost in the conditional central limit theorem for a large class of dependent sequences. Applications to irreducible Markov chains, dynamical systems generated by intermittent maps and $τ$-mixing sequences are given.
We study the Monge and Kantorovich transportation problems on $\mathbb{R}^{\infty}$ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of r…
The duality theory of the Monge-Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let $(\mathcal{X},\mathcal{F},μ)$ and $(\mathcal{Y},\mathcal{G},ν)$ be any probability spaces and $c:\mathcal{X}\times\mathcal{Y}\rightarrow\mathbb{R}$ a measurable cost function such that $f_1+g_1\le c\le f_2+g_2$ for some $f_1,\,f_2\in L_1(μ)$ and $g_1,\,g_2\in L_1(ν)$. Define $α(c)=\inf_P\int c\,dP$ and $α^*(c)=\sup_P\int c\,dP$, where $\inf$ and $\sup$ are over the probabilities $P$ on $\mathcal{F}\otimes\mathcal{G}$ with marginals $μ$ and $ν$. Some duality theorems for…
The de Moivre-Laplace theorem is a special case of the central limit theorem for Bernoulli random variables, and can be proved by direct computation. We deduce the central limit theorem for any random variable with finite variance from the de Moivre-Laplace theorem. Our proof does not use advanced notions such as characteristic functions, the Brownian motion, or stopping times.
This paper aims to establish a central limit theorem for Markov processes conditioned not to be absorbed under a very general assumption on quasi-stationarity for the underlying process. To do so, a central limit theorem has been established for ergodic Markov processes. The conditional central limit theorem is then obtained by applying the central limit theorem to the $Q$-process.
We study optimal mass transport problems between two measures with respect to a non-traditional cost function, i.e. a cost $c$ which can attain the value $+\infty$. We define the notion of $c$-compatibility and strong-$c$-compatibility of two measures, and prove that if there is a finite-cost plan between the measures then the measures must be $c$-compatible, and if in addition the two measures are strongly $c$-compatible, then there is an optimal plan concentrated on a $c$-subgradient of a $c$-class function. This function is the so-called potential of the plan. We give two proofs of this the…
The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes. The central limit theorem and functional central limit theorem are obtained for martingale like random variables under the sub-linear expectation by Zhang (2019). In this paper, we consider the multi-dimensional martingale like random vectors and establish a functional central limit theorem. As applications, the Lindeberg central limit theorem for independent random vectors is established, and the sufficient and necessary conditions of the central limit theorem for independent…
We establish central limit theorems for general functionals on binomial point processes and their Poissonized version. As an application, a central limit theorem for Betti numbers of random geometric complexes in the thermodynamic regime is derived.
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 are natural extension of the classical central limit theory for additive probability measures.
A central limit theorem for arrays of symmetric row-wise exchangeable random variables is presented. The result is valid for finite and infinite extendable and non-extendable sequences. Unlike most reported versions of the central limit theorem valid only for partial sums of finite non-extendable sequences, ours applies to the entire sum and mimics the classical theorem in this sense.
Peng (2006) initiated a new kind of central limit theorem under sub-linear expectations. Song (2017) gave an estimate of the rate of convergence of Peng's central limit theorem. Based on these results, we establish a new kind of almost sure central limit theorem under sub-linear expectations in this paper, which is a quasi sure convergence version of Peng's central limit theorem. Moreover, this result is a natural extension of the classical almost sure central limit theorem to the case where the probability is no longer additive. Meanwhile, we prove a new kind of strong law of large numbers fo…
The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, general central limit theorems and functional central limit theorems are obtained for martingale like random variables under the sub-linear expectation. As applications, the Lindeberg central limit theorem and functional central limit theorem are obtained for independent but not necessarily identically distributed random variables, and a new proof of the Lévy characterization of a G-Brownian motion with…
In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Miłoś (2012, arXiv:1203:6661) and Ren, Song and Zhang (2013, to appear in Acta Appl. Math., DOI 10.1007/s10440-013-9837-0) for supercritical super Ornstein-Uhlenbeck processes. The advantage of this central limit theorem is that it allows us to characterize the limit Gaussian field. In the case of supe…
In this paper we show a central limit theorem for Lebesgue integrals of stationary $BL(θ)$-dependent random fields as the integration domain grows in Van Hove-sense. Our method is to use the (known) analogue result for discrete sums. As applications we obtain various multivariate versions of this central limit theorem.
We prove that the height function associated with the uniform six-vertex model (or equivalently, the uniform homomorphism height function from $\mathbb Z^2$ to $\mathbb Z$) satisfies a central limit theorem, upon some logarithmic rescaling.
Functional central limit theorem for superdiffusive SDEs with stable noise
University of Warwick（华威大学）；Université de Genève, Section de mathématiques（日内瓦大学数学系）；Institute of Mathematics of the National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）；École normale supérieure de Lyon（里昂高等师范学院）
On conditions for Dobrushin's Central limit theorem for non-homogeneous Markov chains
A.A. Kharkevich Institute for Information Transmission Problems of Russian Academy of Sciences（俄罗斯科学院信息传输问题阿·阿·哈尔维奇研究所）；S.M. Nikol’skii Mathematical Institute, RUDN University（俄罗斯人民友谊大学尼古利斯基数学研究所）；National Research University Higher School of Economics（国立高等经济大学）
We investigate a specific infinite urn scheme first considered by Karlin (1967). We prove functional central limit theorems for the total number of urns with at least k balls for different k.
This is a note on some results of the central limit theorem for deterministic dynamical systems. First, we give the central limit theorem for martingales, which is a main tool. Then we give the main results on the central limit theorem in dynamic system in the cases of martingale and backward martingale
For a $d\times d$ expanding matrix $A$, we investigate randomness of the sequence $\{A^k x\}$ and prove the central limit theorem for $\sum f(A^k x)$ where $f$ is a periodic function with a mild regularity condition.
In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren, Song and Zhang[2015]. We first give law of large numbers for supercritical superprocesses with immigration since there is few convergence result on immigration superprocesses, then based on these results, we establish the central limit theorem.
We prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals emerging in stochastic geometry. As a consequence, we provide almost sure central limit theorems for $(i)$ the total edge length of the $k$-nearest neighbors random graph, $(ii)$ the clique count in random geometric graphs, $(iii)$ the volume of the set approximation via the Poisson-Voronoi tessellation.
We explore an asymptotic behavior of Rényi entropy along convolutions in the central limit theorem with respect to the increasing number of i.i.d. summands. In particular, the problem of monotonicity is addressed under suitable moment hypotheses.
