By David Pollard

ISBN-10: 0940600161

ISBN-13: 9780940600164

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**Extra resources for Empirical processes: theory and applications**

**Example text**

4) |f − φi | ≤ α if f ∈ Ei . Each convex combination Σθ(f )f from co(F ) can then be reexpressed as a convex combination of vectors from the convex hulls co(Ei ): θ(f )f = f ∈F λ i ei , i≤m 32 EMPIRICAL PROCESSES where λi = θ(f ) f ∈Ei and ei = f ∈Ei θ(f ) f. λi Here the vector λ of convex weights ranges over the m-dimensional simplex Λ = {λ ∈ Rm : λi ≥ 0 for all i, and λi = 1}. i Because Ei lies inside the unit ball, λ i ei − i≤m |λi − µi |. µi ei ≤ i≤m i≤m We can therefore approximate each point in co(F ) within by means of a convex combination with weights λ chosen from a maximal subset Λ of points from Λ at least apart in 1 distance.

With some adjustment of the constant C after replacement of by /8, this would give the asserted bound for the packing numbers. In what follows the 2 norm will be denoted by | · |, without the subscript 2. Let α = 2/(2 + W ). Choose a maximal subset F of points from F at least apart, then let {φ1 , . . , φm } be a maximal subset of F with points at least α apart. By assumption, m ≤ D2 ( α , F) ≤ A(1/ α )W , #F ≤ D2 ( , F) ≤ A(1/ )W . Notice that m is smaller than A(1/ )τ ; the exponent of 1/ is 2W/(2 + W ), which is less than τ .

Now let T (k) = {t1 , . . , tk } be a ﬁnite set that approximates within a distance δ to every point of T . For a function x in F and a t with ρ(t, ti ) < δ, if x(ti ) < h(ti )−η then x(t) ≤ x(ti ) + η/2 < h(ti ) − η/2. The upper bound is less than h(t), because h ∈ F . A similar argument with g would give a lower bound. It follows that the set {Xn ∈ F : g(ti ) + η < Xn (·, ti ) < h(ti ) − η for ti ∈ T (k)} is contained within {Xn ∈ B}, and hence P∗ {Xn ∈ B} ≥ P{g + η < Xn < h − η on T (k)} − P∗ {Xn ∈ F c }.

### Empirical processes: theory and applications by David Pollard

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