We consider an $L^2$-Wasserstein type distance $ρ$ on the configuration space $Γ_X$ over a Riemannian manifold $X$, and we prove that $ρ$-Lipschitz functions are contained in a Dirichlet space associated with a measure on $Γ_X$ satisfying some general assumptions. These assumptions are in particular fulfilled by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of $ρ$ for a set to be exceptional. This result immediately implies, for instance, a quasi-sure version of the spatial ergodic theorem. We also show that $ρ$ is optimal in the sense that it is the intrinsic metric of our Dirichlet form.
Berry-Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin-Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small-ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.
In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived by Nourdin, Peccati and Reinert(2010) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavin-Stein approach. Besides, we are able to give the almost sure central limit theorem for a sequence of random variables inside a fixed Rademacher chaos using the Ibragimov-Lifshits criterion.
In this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centred Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in Döbler and Krokowski (2017). Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition.
Quantitative multivariate central limit theorems for general functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal approximation. In particular, a discrete multivariate second-order Poincaré inequality is developed. As a first application, the normal approximation of vectors of subgraph counting statistics in the Erdős-Rényi random graph is considered. In this context, we further specialize to the normal approximation of vectors of vertex degrees. In a second application …
Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper extends the classical Menshov--Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate.
We extensively discuss the Rademacher and Sobolev-to-Lipschitz properties for generalized intrinsic distances on strongly local Dirichlet spaces possibly without square field operator. We present many non-smooth and infinite-dimensional examples. As an application, we prove the integral Varadhan short-time asymptotic with respect to a given distance function for a large class of strongly local Dirichlet forms.
Limit theorems for random Dirichlet series with summation over primes, with an application to Rademacher random multiplicative functions
School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv（基辅塔拉斯舍甫琴科国立大学计算机科学与控制论学院）
We study the well-known variational and large deviation principle for graph homomorphisms from $\mathbb{Z}^m$ to $\mathbb{Z}$. We provide a robust method to deduce those principles under minimal a priori assumptions. The only ingredient specific to the model is a discrete Kirszbraun theorem i.e. an extension theorem for graph homomorphisms. All other ingredients are of a general nature not specific to the model. They include elementary combinatorics, the compactness of Lipschitz functions and a simplicial Rademacher theorem. Compared to the literature, our proof does not need any other prelimi…
We consider sub-critical configuration models and show that the central limit theorem for any additive statistic holds when the statistics satisfies a fourth moment assumption, a variance lower bound and the degree sequence of graph satisfies a growth condition. If the degree sequence is bounded, for well known statistics like component counts, log-partition function, and maximum cut-size which are Lipschitz under addition of an edge or switchings then the assumptions reduce to linear growth condition for the variance of the statistic. Our proof is based on an application of the central limit …
An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson's correlation coefficient and the clustering coefficient in the erased configuration …
We consider a stochastic SIR (susceptible $\to$ infective $\to$ recovered) epidemic defined on a configuration model random graph, in which infective individuals can infect only their neighbours in the graph during an infectious period which has an arbitrary but specified distribution. Central limit theorems for the final size (number of initial susceptibles that become infected) of such an epidemic as the population size $n$ tends to infinity, with explicit, easy to compute expressions for the asymptotic variance, are proved assuming that the degrees are bounded. The results are obtained for …
Random Multiplicative Functions and Making Squares from Polynomial Values
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；CNRS UMR 7586 Institut Universitaire de France（法国国家科学研究中心联合研究实验室7586 法国高等科学研究院）；Institut de Mathématiques de Jussieu-Paris Rive Gauche（儒勒·昂里希数学研究所（巴黎左岸））；Institute of Mathematics, Academia Sinica（中央研究院数学研究所）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We study a stochastic compartmental susceptible-infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval $[0,T],$ for some $ T>0$. In this setting, we split the population of graph nodes into two compartments, namely, $S$ and $I$, denoting the susceptible and infected nodes, respectively. In addition to the sizes of these two compartments, we study counts of $SI$-edges (those connecting a susceptible and an infected node) and $SS$-edges (those connecting two susceptible nodes). We describe the dynamical process in terms o…
We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form $T := \sum_{v \in V} H_v$, where $V$ is the vertex set, and $H_v$ depends on a neighbourhood in the graph around $v$ of size at most $\ell$. The error bound is expressed in terms of $\ell$, $|V|$, an almost sure bound on $H_v$, the maximum vertex degree $d_{\max}$ and the variance of $T$. Under suitable assumptions on the convergence of the empirical degree d…
We construct a canonical differential structure on the configuration space $Υ$ over a singular base space $X$ and with a general invariant measure $μ$ on $Υ$. We present an analytic structure on $Υ$, constructing a strongly local Dirichlet form $\mathcal E$ on $L^2(Υ, μ)$ for $μ$ in a large class of probability measures. We then investigate the geometric structure of the extended metric measure space $Υ$ endowed with the $L^2$-transportation extended distance $\mathsf{d}_Υ$ and with the measure $μ$. By establishing Rademacher- and Sobolev-to-Lipschitz-type properties for $\mathcal E$, we final…
