### A general version of delta theorem

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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.

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.

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...