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By Walter Philipp

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SECTION 6 Convex Hulls Sometimes interesting random processes are expressible as convex combinations of more basic processes. For example, if 0 ≤ fi ≤ 1 for each i then the study of fi reduces to the study of the random sets {ω : s ≤ fi (ω, t)}, for 0 ≤ s ≤ 1 and t ∈ T , by means of the representation 1 {s ≤ fi (ω, t)} ds. fi (ω, t) = 0 More generally, starting from fi (ω, t) indexed by T , we can construct new processes by averaging out over the parameter with respect to a probability measure Q on T : fi (ω, Q) = fi (ω, t)Q(dt).

The expectation with respect to Pσ on the right-hand side of the last expression is less than n |Fn |1 + 1 Pσ max |σ · f ∗ |. n Dnω The first of these terms has a small expectation, because assumption (i) implies uniform boundedness of n1 P|Fn |1 . The second term is bounded by K. 1) it is also less than C max |f ∗ |2 n Dnω 2 + 2 log Mn . √ The square root √ factor contributes at most op ( n) to this bound. The other factor is of order Op ( n), because, for each point in Fnω , |f ∗ |22 = fi2 {Fi ≤ K} ≤ K i≤n Fi .

Dudley (1985) has shown that the Donsker-class property is preserved under the formation of (sequential closures of) convex hulls of classes of functions. ) This gives yet another way of handling processes representable as convex combinations of simpler processes. The same stability property is also implied by the first theorem of Talagrand (1987). SECTION 7 Maximal Inequalities Let us now pull together the ideas from previous sections to establish a few useful maximal inequalities for the partial-sum process Sn .

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