By Walter Philipp

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Additional info for Almost sure invariance principles for partial sums of weakly dependent random variables

Example text

It suffices to show that card{j S n: Vj E R'} ~ card(I). Since the points Vj are spread uniformly, it suffices to show that card(I) S n D. A square of G~ in R' \ R is (lRI + ~ IR' \ RI) IDI denotes the area of connex components of R. ) Thus, it suffices to show that for R connex card(I) S n (IRI + 1161R, \ RI) . Taking differences one sees that it suffices to show that for the interior 6 of a simple curve C where R' (resp. R") is the union of the squares of G~ that touch [0,1]2 \ R'). Now 16\ Rill ~ 2- t ,-2P(C) and IR' \ 61 ~ 6 (resp.

2) holds whenever both: (I) The class C, considered as a subset of L2, is small; and (II) du(·,·) and 11·112 are well comparable on C. Property (I) is quantified by the existence of certain majorizing measures on C. Checking this is certainly the difficult part, and in the case of Shor's Discrepancy Theorem goes beyond the scope of these notes. Although property (II) is easier with which to deal, it imposes more restrictions on C than property (I). For example, for each 1 < Q :::; 00 the class C(Q)={f:[0,1]2->R: J{ lrD,I]' f(x,y) dxdy = 0, lI~flh:::;l, II~fll ..

Matching random samples in dimension 3 (or more). Forthcoming. [18] Talagrand, M. (1992c). Matching random samples in many dimensions. To appear in Ann. Appl. Probab. [19] Yukich, J. (1992a). The exponential integrability of transportation cost. Preprint. [20] Yukich, J. (1992b). Some generalizations of the Euclidean two-sample matching problem. In this volume. Marjorie G. Hahn and Yongzhao Shao Department of Mathematics Tufts University Medford, MA 02155 USA THE AJTAI-KOMLOS-TUSNADY MATCHING THEOREM FOR GENERAL MEASURES Michel Talagrand(*) Abstract.