Donsker classes of sets

Summary We study the central limit theorem (CLT) and the law of large numbers (LLN) for empirical processes indexed by a (countable) class of sets C. The main result, of purely measure-theoretical nature, relates different ways to measure the size of C. It relies on a new rearrangement inequality that has been inspired by techniques used in the local theory of Banach spaces. As an application, we give sharp necessary conditions for the CLT, that are in some sense the best possible. We also obtain a way to compute the rate of convergence in the LLN.     

A general form of certain mean field models for spin glasses

Given numbers a _ij ≥ 0 for 1 ≤ i  <  j ≤ N, and given numbers b _i ≥ 0, i ≤ N, we consider the random Hamiltonian $\sum_{i,j \le N} \sqrt{a_{ij}} g_{ij} \sigma_i \sigma_j + \sum_{i \le N} \sqrt{b_i} g_i \sigma_i$ , where g _i , g _ij denote independent standard normal r.v., and where σ _i = ± 1. We give sufficient conditions on the coefficients a _ij for the system governed by this Hamiltonian to exhibit “high-temperature behavior”. There results extend known facts concerning the behavior of the Sherrington-Kirkpatrick model at “very high-temperature”. In a similar manner we give a general form of the “perceptron model”.     

Lower classes for fractional Brownian motion

We characterize the lower classes of fractional Brownian motion by an integral test.Work partially supported by an NSF grant. Equipe d'Analyse, Tour 46, U.A. at C.N.R.S. n^o 754, Université Paris VI, 4 place Jussieu, 75230 Paris Cedex 05, and Department of Mathematics, 231 West 18th Avenue, Columbus, Ohio 43210.     

Expected wasted space of optimal simple rectangle packing

A simple packing of a collection of rectangles contained in [0,1]² is a disjoint subcollection such that each vertical line meets at most one rectangle of the packing. The wasted space of the packing is the surface of the area of the part of [0,1]² not covered by the packing. We prove that for a certain number L, for all N2, the wasted space W_N in an optimal simple packing of N independent uniformly distributed rectangles satisfiesWork partially supported by an N.S.F. grant.Mathematics Subject Classification (2000): 60D05     

A new countably determined Banach space

We construct a Banach space which is weak^*-countably determined in its second dual, but which is notK-analytic for its weak topology. This paper was written while the author was visiting The Ohio State University.     

Separabilite vague dans l’espace des mesures sur un compact

Using the continuum hypothesis we construct a compact spaceK such that the spaceM (K)) of measures onK is vaguely separable, i.e., thatC (K) is injected intol ^∞, but thatC (K) is not isomorphic to a subspace ofl ^∞. It is shown that ifC(K) is isomorphic to a subspace ofC (K) is positively isometric to a subspace ofl ^∞(⌈). Nevertheless, under the continuum hypothesis one can construct a compact spaceL such that the spaceM ₁ ⁺ (L) of probabilities onL is vaguely separable, butL cannot be the support of a measureμ withL ¹(μ) separable in the norm.      

Selector processes on classes of sets

Consider the random subset X of ℕ obtained by selecting independently each integer with a probability δ. Consider a finite class of finite sets. We describe a combinatorial quantity that is of the same order as We then give a related result allowing to compute the supremum of the empirical process on a class of sets. Work partially supported by an NSF grant.     

On the meaning of Parisi's functional order parameter

The author has recently proved that a famous formula discovered by G. Parisi gives at any temperature the correct value for the limiting free energy of a large class of mean field models for spin glasses (a class which contains in particular the Sherrington–Kirkpatrick model). Here we prove rigorously that (generically) the “functional order parameter” occuring in this formula can be interpreted as predicted by Parisi, namely as representing the limiting distribution of the overlap of two independent configurations. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).