controls terms of the type appearing in C2). In Appendix B.1 we restate specific results from these authors which are useful to study C2). Finally, the following lemma gives sufficient conditions under which C1) holds. We first set K$_{1}$$_{n}$ ≜ ‖ L RS ˆ P$_{n}$ ‖$_{1}$$_{,P}$$_{0}$ + ‖ M RS ˆ P$_{n}$ ‖ 2 2 $_{,P$_{0}$}$. Lemma 1 (Sufficient conditions for C1)) . Suppose that o$_{1}$ ↦→ ∫ Γ$_{n}$( o$_{1}$,o ) dP$_{0}$ ( o ) /K$_{1}$$_{n}$ , defined to be zero if K$_{1}$$_{n}$ = 0 , belongs to a P$_{0}$ -Donsker class (van der Vaart and Wellner, 1996) with probability tending to 1 . Then, under H$_{0}$ , and thus C1) holds whenever K$_{1}$$_{n}$ = o$_{P}$$_{0}$ ( n − 1 / $^{2}$) . The following theorem describes the asymptotic distribution of nψ$_{n}$ under the null hy- pothesis whenever conditions C1), C2) and C3) are satisfied. Theorem 4 (Asymptotic distribution under H$_{0}$ ) . Suppose that C1), C2) and C3) hold. Then, under H$_{0}$ , where { λ$_{k}$ } ∞ k =1 are the eigenvalues of the integral operator h ( o ) ↦→ ∫ Γ$_{0}$( o$_{1}$,o ) h ( o ) dP$_{0}$ ( o$_{1}$ ) repeated according to their multiplicity, and { Z$_{k}$ } ∞ k =1 is a sequence of independent standard normal random variables. Furthermore, all of these eigenvalues are nonnegative under H$_{0}$ . We note that by employing a sample splitting procedure – namely, estimating Γ$_{0}$ on one portion of the sample and constructing the U -statistic based on the remainder of the sample – it is possible to eliminate the U -process conditions required for C2). In such a case, satisfaction of C2) only requires convergence of ˜ Γ$_{n}$ to Γ$_{0}$ with respect to the L $^{2}$( P 2 $_{0}$)-norm. Remark 3. In Example 3, sample splitting may prove particularly important when the es- timator of E$_{P}$$_{0}$ ( Y | W = w ) is chosen as the minimizer of an empirical risk since in finite samples the bias induced by using the same residuals y − E$_{ˆ}$ P$_{n}$ ( Y | W = w ) as those in the 16 nψ$_{n}$ = n$_{U}$$_{n}$ Γ$_{0}$ + o$_{P}$$_{0}$ (1) ⇝ ∞ ∑ k =1 λ$_{k}$ $^{(}$Z 2 k − 1 $^{)}$, ( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{n}$ = O$_{P}$$_{0}$$^{(}$ K$_{1}$$_{n}$ √ n$^{)}$