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. 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 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): 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- 39 ∫ ∞ 0 sup Q$_{1}$ $^{√}$log N ( ϵ, Q$_{1}$, G$_{1}$ , G$_{1}$ ) dϵ ≤ ∫ ∞ 0 sup Q$_{2}$ $^{√}$log N ( ϵ, Q$_{2}$, G$_{2}$ , G$_{2}$ ) dϵ, (A.3) sup Q$_{1}$ N ( ϵ, Q$_{1}$, G$_{1}$ , G$_{1}$ ) ≤ sup Q$_{2}$ N ( ϵ, Q$_{2}$, G$_{2}$ , G$_{2}$ ) , (A.2) Appendix B.2: Controlling ∫ Γ$_{n}$( · , o ) dP$_{0}$ ( o )