properties discussed in this appendix are equivalent for any set of functions h$_{1}$ that yield the same tangent space. Hence, the restriction that sup$_{o}$$_{∈O}$ | h$_{1}$ ( o ) | < ∞ , while convenient, will have no impact on the resulting differentiability properties. The second-order tangent space is also determined by the first-order tangent space (see Carone et al., 2014 and the references therein). Let ψ$_{t}$ ≜ Ψ( P$_{t}$ ). The parameter Ψ is called (first-order) pathwise differentiable at P$_{0}$ if there exists a D Ψ 1 ∈ L 2 $_{0}$( P$_{0}$ ) such that We call D Ψ 1 the first-order canonical gradient of Ψ at P$_{0}$ , where we note that D Ψ 1 ( O ) is almost surely unique because M is nonparametric. The canonical gradient D Ψ 1 depends on P$_{0}$ but this is omitted in the notation because we will only discuss pathwise differentiability at P$_{0}$ . A function f : O 2 →$_{R}$ is called ( P ) one-degenerate if it is symmetric and Pf ( o, · ) = 0. We will use the notation P $^{2}$f = E$_{P}$ 2 [ f ( O$_{1}$, O$_{2}$ )]. The parameter Ψ is called second-order pathwise differentiable at P$_{0}$ if there exists some symmetric, one-degenerate, P 2 0 square integrable function D Ψ 2 such that We now present two results from empirical process theory, the first of which can be used to control the U -processes that we deal with in the main text when H$_{0}$ holds, and the second of which can be used to establish an empirical process condition that is used when H$_{1}$ holds. Before giving an overview of the empirical process theory that we use, we review the notion of a covering number. Let µ be a probability measure over Z . For a class of functions f : Z →$_{R}$ with envelope F (i.e., | f ( z ) | ≤ F ( z ) for all z ∈ Z ), where 0 < ‖ F ‖$_{2}$$_{,µ}$ < ∞ , define 36 ψ$_{t}$ − ψ$_{0}$ = tP$_{0}$D Ψ $_{1}$h$_{1}$ + 1 2 t $^{2}$P$_{0}$ D Ψ $_{1}$h$_{2}$ + 1 2 t 2 ∫ ∫ D Ψ 2 ( o$_{1}$, o$_{2}$ ) h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) + o ( t $^{2}$) . Appendix B: Empirical Process Results ψ$_{t}$ − ψ$_{0}$ = tP$_{0}$D Ψ $_{1}$h$_{1}$ + o ( t ) .