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Vaart and Wellner, 1996). 

For each g ∈ G , let H$_{g}$ represent the Hilbert-Schmidt operator on L $^{2}$( P$_{0}$ ) given by ( H$_{g}$f )( o ) = Pg ( o, · ) f ( · ). Let { W$_{j}$ : j = 1 , 2 , ... } be a sequence of i.i.d. standard normal random variables and { w$_{j}$ : j = 1 , 2 , ... } be an orthonormal basis of L $^{2}$( P$_{0}$ ). Let ˜ Q be a process on G defined by 

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A functional M : G →$_{R}$ is said to belong to C ( G , P 2 $_{0}$) if g ↦→ M ( g ) is uniformly continuous for the L $^{2}$( P 2 $_{0}$) seminorm and sup$_{G}$ | M ( g ) | < ∞ . 

We have modified the statement from Nolan and Pollard (1988) slightly to apply to the entropy integral given in (A.1). We omit an analogue to condition (ii) from Nolan and Pollard’s statement of the theorem below because it is implied by our strengthening of their condition (i). 

Theorem A.1 (Theorem 7, Nolan and Pollard, 1988) . Suppose that the one-degenerate class G satisfies 

(i’) J (1 , G , G ) < ∞ ; 

(iii’) sup$_{Q}$ log N ( ϵ, Q × P$_{0}$, G , G ) < ∞ for each ϵ > 0 , where the supremum is over distribu- tions Q with support O . 

Then there is a version of ˜ Q with continuous sample paths in C ( G , P$_{0}$ × P$_{0}$ ) and n$_{U}$$_{n}$ converges in distribution to ˜ Q . 

We will use the following corollary to control the cross-terms. 

Corollary A.1. Suppose that G satisfies the conditions of A.1 and ˆ g$_{n}$ is a sequence of one- degenerate random functions that take their values in G such that P 2 $_{0}$(ˆ g$_{n}$ − g$_{0}$ ) 2 → 0 in probability for some g$_{0}$ ∈ G . Then n$_{U}$$_{n}$ (ˆ g$_{n}$ − g$_{0}$ ) → 0 in probability. 

