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We give an exact solution for the complete distribution of component sizes in random networks with arbitrary degree distributions. The solution tells us the probability that a randomly chosen node belongs to a component of size s, for any…
We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge…
We theoretically investigate the correlation functions of the phase of a light wave propagating through a turbulent medium. We use an equation for the logarithm of a wave packet envelope, which includes a second-order nonlinear term. Based…
This paper presents a data-driven framework for uncertainty propagation under unmeasured or statistically unmodeled (unstructured) disturbances. We consider residual disturbances, which consolidate all unstructured disturbances into a…
An equilibrium statistical system is known to be stable if the fluctuations of global observables are normal, when their dispersions are proportional to the number of particles, or to the system volume. A general theorem is rigorously…
One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams. Despite much effort, this is still largely an open problem. In this paper, we present a series of…
Random forests remain among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe their performance and…
We establish a link between the phenomenon of Taylor dispersion and the theory of empirical distributions. Using this connection, we derive, upon applying the theory of large deviations, an alternative and much more precise description 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…
Researchers often use linear regression to analyse randomized experiments to improve treatment effect estimation by adjusting for imbalances of covariates in the treatment and control groups. Our work offers a randomization-based inference…
The classical fluctuation-dissipation theorem predicts the average response of a dynamical system to an external deterministic perturbation via time-lagged statistical correlation functions of the corresponding unperturbed system. In this…
In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the mathematical characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical…
Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the…
We analyze the fluctuations of incomplete $U$-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled…
We examine the problem of family size statistics (the number of individuals carrying the same surname, or the same DNA sequence) in a given size subsample of an exponentially growing population. We approach the problem from two directions.…
A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the…
We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour spreads out trough an infinite number of individuals. We present sharp lower…
In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic…
We investigate the asymptotic behavior of posterior distributions of regression coefficients in high-dimensional linear models as the number of dimensions grows with the number of observations. We show that the posterior distribution…
In this article, we try to give an answer to the simple question: ``\textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection…