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A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes

Dietmar Ferger (2021)

Kybernetika

For lower-semicontinuous and convex stochastic processes Z n and nonnegative random variables ϵ n we investigate the pertaining random sets A ( Z n , ϵ n ) of all ϵ n -approximating minimizers of Z n . It is shown that, if the finite dimensional distributions of the Z n converge to some Z and if the ϵ n converge in probability to some constant c , then the A ( Z n , ϵ n ) converge in distribution to A ( Z , c ) in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in...

A remark on Vapnik-Chervonienkis classes

Agata Smoktunowicz (1997)

Colloquium Mathematicae

We show that the family of all lines in the plane which is a VC class of index 2 cannot be obtained in a finite number of steps starting with VC classes of index 1 and applying the operations of intersection and union. This confirms a common belief among specialists and solves a question asked by several authors.

A two armed bandit type problem revisited

Gilles Pagès (2005)

ESAIM: Probability and Statistics

In Benaïm and Ben Arous (2003) is solved a multi-armed bandit problem arising in the theory of learning in games. We propose a short and elementary proof of this result based on a variant of the Kronecker lemma.

A two armed bandit type problem revisited

Gilles Pagès (2010)

ESAIM: Probability and Statistics

In Benaïm and Ben Arous (2003) is solved a multi-armed bandit problem arising in the theory of learning in games. We propose a short and elementary proof of this result based on a variant of the Kronecker lemma.

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Antoine Gloria, Stefan Neukamm, Felix Otto (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm...

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