By David Pollard

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

**Sample text**

To accommodate a wider range of applications, let us expand the setting to cover triangular arrays of random processes, {fni (ω, t) : t ∈ T, 1 ≤ i ≤ kn } for n = 1, 2, . . , independent within each row. To facilitate application of the stability arguments, let us also allow for nonnegative rescaling vectors. 9) Definition. Call a triangular array of processes {fni (ω, t)} manageable (with respect to the envelopes Fn (ω)) if there exists a deterministic function λ, the capacity bound , for which 1 (i) 0 log λ(x) dx < ∞, (ii) D(x|α Fn (ω)|, α Fnω ) ≤ λ(x) for 0 < x ≤ 1, all ω, all vectors α of nonnegative weights, and all n.

Argue from convexity of Φ and the inequality fi+ = 1/2(fi + |fi |) that Pσ Φ 2 sup F σi fi+ ≤ i≤n 1 Pσ Φ 2 sup 2 F σi fi i≤n 1 + Pσ Φ 2 sup 2 F σi |fi | , i≤n with a similar inequality for the contribution from the −fi− term. The proof will be completed by an application of the Basic Combinatorial Lemma from Section 1 to show that Pσ Φ 2 sup F σi |fi | ≤ 2 Pσ Φ 2 sup F i≤n σi fi . i≤n Because Φ is increasing and nonnegative, and F contains the zero vector, Pσ Φ 2 sup F σi |fi | ≤ Pσ Φ 2 sup F i≤n i≤n σi |fi | + Pσ Φ 2 sup F i≤n (−σi )|fi | .

CONVERGENCE IN DISTRIBUTION 45 It has long been recognized (see Pyke 1969, for example) that many arguments involving convergence in distribution are greatly simpliﬁed by use of a technical device known as almost sure representation. Such a representation usually asserts something like: If Xn P then there exist Xn and X such that Xn and Xn have the same distribution, X has distribution P , and Xn → X almost surely. The random elements Xn and X are deﬁned on a new probability space (Ω, A, P). For Borel measurable Xn , “the same distribution” is interpreted to mean that Pg(Xn ) = Pg(Xn ) for all bounded, Borel measurable g.