Product measure
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Monte Carlo with determinantal point processes Open
We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures …
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Monotone Cellular Automata in a Random Environment Open
In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in $\mathbb{Z}$ d with random initial configurations. Formally, we are given a set $\mathcal{U}$ = { X 1 ,…
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Reproducing pairs of measurable functions and partial inner product spaces Open
We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product spac…
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Generalization Error Bounds Via Rényi-, $f$-Divergences and Maximal Leakage Open
In this work, the probability of an event under some joint distribution is bounded by measuring it with the product of the marginals instead (which is typically easier to analyze) together with a measure of the dependence between the two r…
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Quantum and spectral properties of the Labyrinth model Open
We consider the Labyrinth model, which is a two-dimensional quasicrystal model. We show that the spectrum of this model, which is known to be a product of two Cantor sets, is an interval for small values of the coupling constant. We also c…
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Hall-Littlewood-PushTASEP and its KPZ limit Open
We study a new model of interactive particle systems which we call the randomly activated cascading exclusion process (RACEP). Particles wake up according to exponential clocks and then take a geometric number of steps. If another particle…
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An information-theoretic proof of a finite de Finetti theorem Open
A finite form of de Finetti’s representation theorem is established using elementary information-theoretic tools: The distribution of the first k random variables in an exchangeable binary vector of length n ≥ k is close to a mixture of pr…
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Concentration of Product Spaces Open
We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also conce…
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On a class of Hausdorff measure of cartesian product sets in metric spaces Open
In this paper we study, in a separable metric space, a class of Hausdorff measures ${\mathcal H}_\mu^{q, \xi}$ defined using a measure $\mu$ and a premeasure $\xi$. We discuss a Hausdorff structure of product sets. Weighted Hausdorff measu…
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Construction of Fuzzy Measures over Product Spaces Open
In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of …
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Topics of Measure Theory on Infinite Dimensional Spaces Open
This short review is devoted to measures on infinite dimensional spaces. We start by discussing product measures and projective techniques. Special attention is paid to measures on linear spaces, and in particular to Gaussian measures. Tra…
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Fubini’s Theorem on Measure Open
Summary The purpose of this article is to show Fubini’s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of a…
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The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Process Open
We describe the translation invariant stationary states of the one dimensional discrete-time facilitated totally asymmetric simple exclusion process (F-TASEP). In this system a particle at site $j$ in $Z$ jumps, at integer times, to site $…
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Product Pre-Measure Open
Summary In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existe…
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A new look at random projections of the cube and general product measures Open
A strong law of large numbers for $d$-dimensional random projections of the $n$-dimensional cube is derived. It shows that with respect to the Hausdorff distance a properly normalized random projection of $[-1,1]^n$ onto $\mathbb{R}^d$ alm…
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The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations Open
The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $π$ is proportional to $q^{l(π)}$ with $q > 0$ being a parameter and $l(π)$ the number of inversions in $π$. We show the convergence of the random…
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Product of Measurable Spaces and Applications Open
We deal with products of measurable spaces and relationships between measures on products and asymmetrical stochastic dependence/independence of one extended probability space on another one.
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On Pointwise Product Vector Measure Duality Open
This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral…
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Multidimensional Measure Space and Integration Open
Summary This paper introduces multidimensional measure spaces and the integration of functions on these spaces in Mizar. Integrals on the multidimensional Cartesian product measure space are defined and appropriate formal apparatus to deal…
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Concentration of Product Spaces Open
We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also conce…
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Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions Open
The seminal result of Kahn, Kalai and Linial shows that a coalition of O(n/(log n)) players can bias the outcome of any Boolean function {0,1}^n -> {0,1} with respect to the uniform measure. We extend their result to arbitrary product meas…
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Exact Solution of the F-TASEP Open
We obtain the exact solution of the facilitated totally asymmetric simple exclusion process (F-TASEP) in 1D. The model is closely related to the conserved lattice gas (CLG) model and to some cellular automaton traffic models. In the F-TASE…
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Overlapping Iterated Function Systems from the Perspective of Metric Number Theory Open
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their L…
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Product Measure Approximation of Symmetric Graph Properties Open
In the study of random structures we often face a trade-off between realism and tractability, the latter typically enabled by assuming some form of independence. In this work we initiate an effort to bridge this gap by developing tools tha…
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When does the chaos in the Curie-Weiss model stop to propagate? Open
We investigate increasing propagation of chaos for the mean-field Ising model of ferromagnetism (also known as the Curie-Weiss model) with N spins at inverse temperature β>0 and subject to an external magnetic field of strength h∈R. Using …
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Product Mixing in Compact Lie Groups Open
If G is a group, we say a subset S of G is product-free if the equation xy=z has no solutions with x,y,z ∈ S.In 1985, Babai and Sós [] asked, for a finite group G, how large a subset S⊆ G can be if it is product-free. The main tool (hither…
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The semigroup of metric measure spaces and its infinitely divisible probability measures Open
A metric measure space is a complete, separable metric space equipped with a probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure…
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From the divergence between two measures to the shortest path between two observables Open
We consider two independent and stationary measures over $\unicode[STIX]{x1D712}^{\mathbb{N}}$ , where $\unicode[STIX]{x1D712}$ is a finite or countable alphabet. For each pair of $n$ -strings in the product space we define $T_{n}^{(2)}$ a…
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Threshold for the expected measure of the convex hull of random points with independent coordinates Open
Let be an even Borel probability measure on . For every , consider independent random vectors in , with independent coordinates having distribution . We establish a sharp threshold for the product measure of the random polytope in under th…
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Coboundaries of nonconventional ergodic averages Open
Let $(X,\mathcal{A}, μ)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be invertible bi measurable measure preserving transformations on this measure space. We give a sufficient condition for the product of $H$ bounded fun…