Related papers: Push forward measures and concentration phenomena

For $\ell\colon \mathbb{R}^d \to [0,\infty)$ we consider the sequence of probability measures $\left(\mu_n\right)_{n \in \mathbb{N}}$, where $\mu_n$ is determined by a density that is proportional to $\exp(-n\ell)$. We allow for infinitely…

For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E}…

Let $\nu$ be a Borel probability measure on a $d$-dimensional Euclidean space $\mathbb{R}^d$, $d\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \ldots, p_N)$ be a probability vector with $p_j>0$ for $0\leq j\leq N$. Let $\{S_j:…

In a Bayesian inverse problem setting, the solution consists of a posterior measure obtained by combining prior belief, information about the forward operator, and noisy observational data. This measure is most often given in terms of a…

We utilize the concentration of measure phenomenon to study the large $N$ limit of the $O(N)$ principal chiral model. The partition function in this limit is demonstrated to be that of a free massive theory.

We show that there is a sharp threshold in dimension one for the transport cost between the Lebesgue measure $\lambda$ and an invariant random measure $\mu$ of unit intensity to be finite. We show that for \emph{any} such random measure the…

This article deals with Coulomb gases at an intermediate temperature regime, in which no structure is observed at the microscopic level, but the mass in confined to a compact set. Our main result is a concentration inequality around the…

The present work provides an original framework for random matrix analysis based on revisiting the concentration of measure theory from a probabilistic point of view. By providing various notions of vector concentration ($q$-exponential,…

We consider a random variable $X$ that takes values in a (possibly infinite-dimensional) topological vector space $\mathcal{X}$. We show that, with respect to an appropriate "normal distance" on $\mathcal{X}$, concentration inequalities for…

We introduce a sharpness functional for probabilistic models that quantifies sharpness as an intrinsic property of the probability distribution. The measure is derived based on a rank-based concentration principle that tracks upward…

We provide an algorithm to approximate a finitely supported discrete measure $\mu$ by a measure $\nu_{N}$ corresponding to a set of $N$ points so that the total variation between $\mu$ and $\nu_N$ has an upper bound. As a consequence if…

Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the…

Suppose that two large, multi-dimensional data sets are each noisy measurements of the same underlying random process, and principle components analysis is performed separately on the data sets to reduce their dimensionality. In some…

It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze…

Let $M$ be a complete Riemannian manifold, $N\in \NN$ and $p\ge 1$. We prove that almost everywhere on $x=(x_1,...,x_N)\in M^N$ for Lebesgue measure in $M^N$, the measure $\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$ has a unique $p$-mean $e_p(x)$.…

Choosing models from a hypothesis space is a frequent task in approximation theory and inverse problems. Cross-validation is a classical tool in the learner's repertoire to compare the goodness of fit for different reconstruction models.…

This paper deals with studying vague convergence of random measures of the form $\mu_{n}=\sum_{i=1}^{n} p_{i,n} \delta_{\theta_i}$, where $(\theta_i)_{1\le i \le n}$ is a sequence of independent and identically distributed random variables…

This work addresses potential theoretic questions for the standard nearest neighbor random walk on the hypercube $\{-1,+1\}^N$. For a large class of subsets $A\subset\{-1,+1\}^N$ we give precise estimates for the harmonic measure of $A$,…

Let $\mu$ be an even Borel probability measure on ${\mathbb R}$. For every $N>n$ consider $N$ independent random vectors $\vec{X}_1,\ldots ,\vec{X}_N$ in ${\mathbb R}^n$, with independent coordinates having distribution $\mu $. We establish…

Given a bounded operator $T$ on a Banach space $X$, we study the existence of a probability measure $\mu$ on $X$ such that, for many functions $f:X\to\mathbb K$, the sequence $(f+\dots+f\circ T^{n-1})/\sqrt n$ converges in distribution to a…