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A uniform central limit theorem for dependent variables

Konrad Furmańczyk (2009)

Applicationes Mathematicae

Niemiro and Zieliński (2007) have recently obtained uniform asymptotic normality for the Bernoulli scheme. This paper concerns a similar problem. We show the uniform central limit theorem for a sequence of stationary random variables.

Affine Dunkl processes of type A ˜ 1

François Chapon (2012)

Annales de l'I.H.P. Probabilités et statistiques

We introduce the analogue of Dunkl processes in the case of an affine root system of type A ˜ 1 . The construction of the affine Dunkl process is achieved by a skew-product decomposition by means of its radial part and a jump process on the affine Weyl group, where the radial part of the affine Dunkl process is given by a Gaussian process on the ultraspherical hypergroup [ 0 , 1 ] . We prove that the affine Dunkl process is a càdlàg Markov process as well as a local martingale, study its jumps, and give a martingale...

An isomorphic Dvoretzky's theorem for convex bodies

Y. Gordon, O. Guédon, M. Meyer (1998)

Studia Mathematica

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in n with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of n satisfying d ( Y K , B 2 k ) C ( 1 + ( k / l n ( n / ( k l n ( n + 1 ) ) ) ) . This formulation of Dvoretzky’s theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex.

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