content_image/314924.jpg content_image/314923.jpg
The proof relies on the continuity of sample paths of (a version of) ˜ Q . The proof is omitted, but we refer the reader to the proof of Lemma 19.24 in van der Vaart (1998) for the analogous empirical process result. 

## Appendix B.2: Controlling ∫ Γ$_{n}$( · , o ) dP$_{0}$ ( o ) 

We now give sufficient conditions under which o ↦→ ∫ Γ$_{n}$( o$_{1}$, o ) dP$_{0}$ ( o ) belongs to a fixed Donsker class with probability approaching one. We recall from van der Vaart and Wellner (1996) that a class F of functions mapping from O to R is Donsker if its uniform entropy integral is finite, which holds if its covering number grows sufficiently slowly as the approxi- mation becomes arbitrarily precise. 

Let G$_{2}$ be some class of functions g : O 2 → [ − M, M ], M < ∞ , that contains { ( o$_{1}$, o$_{2}$ ) ↦→ Γ$_{n}$( o$_{1}$, o$_{2}$ ) : Γ$_{n}$ } . Without loss of generality, we suppose that M = 1. We take the constant function G$_{2}$ ≡ 1 as envelope for G$_{2}$ . Let G$_{1}$ ≜ { o$_{1}$ ↦→ ∫ g$_{2}$ ( o$_{1}$, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) : g$_{2}$ ∈ G$_{2}$} , and note that this class similarly has envelope G$_{1}$ ≡ 1. The main observation of this subappendix is that 

<img src='content_image/314924.jpg'>

where the supremum on the left is over all distributions Q$_{1}$ on O such that ‖ G$_{1}$ ‖$_{2}$$_{,Q}$$_{1}$ > 0 and the supremum on the right is over all distributions Q$_{2}$ on O 2 such that ‖ G ‖$_{2}$$_{,Q}$$_{2}$ > 0. If we can show this, then the uniform entropy integrals are also ordered (Section 2.5 in van der Vaart and Wellner, 1996): 

<img src='content_image/314923.jpg'>

where the left- and right-hand sides are the uniform entropy integrals of G$_{1}$ and G$_{2}$ , re- spectively. Hence, it will suffice to show that the right-hand side is finite. This can be accomplished using the variety of techniques given in Chapter 2 of van der Vaart and Well- 

