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The ratio between the probability that two distributions $R$ and $P$ give to points $x$ are known as importance weights or propensity scores and play a fundamental role in many different fields, most notably, statistics and machine…
Let $M_2$ be the space of quadratic rational maps $f:{\bf P}^1\to{\bf P}^1$, modulo the action by conjugation of the group of M\"obius transformations. In this paper a compactification $X$ of $M_2$ is defined, as a modification of Milnor's…
Let (S(t)) be a one-parameter family S = (S(t)) of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the transition…
We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using Transportation Cost Inequalities for stochastic…
We discuss the spectrum of the tensor metric perturbations and the stability of warped compactifications with the de Sitter spacetime in the higher-dimensional gravity. The spacetime structure is given in terms of the warped product of the…
Overconfidence and underconfidence in machine learning classifiers is measured by calibration: the degree to which the probabilities predicted for each class match the accuracy of the classifier on that prediction. How one measures…
We consider the Buhmann class of compactly supported radial basis functions, whih includes a wealth of special cases that have been studied in both numerical analysis and spatial statistics literatures. In particular, the celebrated Wu,…
Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly…
We study the large time behavior of the optimal transportation cost towards the uniform distribution, for the occupation measure of a stationary Brownian motion on the flat torus in $d$ dimensions, where the cost of transporting a unit of…
We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $\mathbb {R}^d$ under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under…
Let $\rho$ be compactly supported on $D \subset \mathbb R^2$. Endow $\mathbb R^2$ with the metric $e^{\rho}(dx_1^2 + dx_2^2)$. As $\delta \to 0$ the set of Brownian loops centered in $D$ with length at least $\delta$ has measure…
The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These…
We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the…
A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts,…
We prove that in any finite set of $\mathbb Z^d$ with $d\ge 3$, there is a subset whose capacity and volume are both of the same order as the capacity of the initial set. As an application we obtain estimates on the probability of {\it…
We discuss the compact support property of the rough super-Brownian motion constructed as a scaling limit of a branching random walk in static random environment. The semi-linear equation corresponding to this measure-valued process is the…
Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with $L_2$ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong…
In compact settings, the convergence rate of the empirical optimal transport cost to its population value is well understood for a wide class of spaces and cost functions. In unbounded settings, however, hitherto available results require…
It was proved in 1998 by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the…
We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a…