function Q$_{ψ}$ is continuous in its arguments, by Donsker’s theorem we can express the limit process for √$_{n}$ ( ˆ F$_{y}$$_{|}$$_{x}$ − F$_{y}$$_{|}$$_{x}$, ψ ) as ( Q$_{y}$$_{|}$$_{x}$, ψ ) = Q$_{ψ}$ ( U$_{x}$, U$_{xy}$ ) by substituting the limit Browning bridge processes for the arguments of Q$_{ψ}$ ( ., . ) . For any ψ$_{1}$, ..., ψ$_{l}$ ∈ D ( W ) the joint limit process for is similarly given by the joint process of Q$_{ψ}$$_{1}$ ( U$_{x}$, U$_{xy}$ ) , ..., Q$_{ψ}$$_{l}$ ( U$_{x}$, U$_{xy}$ ) . This is a Gaussian process. The mean is zero since Q$_{ψ}$ is linear in its argu- ments and the covariance is given by cov ( Q$_{ψ}$$_{1}$ ( U$_{x}$, U$_{xy}$ ) , Q$_{ψ}$$_{2}$ ( U$_{x}$, U$_{xy}$ ) ) = cov ( ( Q$_{y}$$_{|}$$_{x}$, ψ$_{1}$ ) , ( Q$_{y}$$_{|}$$_{x}$, ψ$_{2}$ ) ) . Existence follows from boundedness of the func- tions in the expressions and bounded support of ψ. By assumption of the theorem h 2 = o ( n − 1 2 ) , thus the limit process is fully described by Q$_{y}$$_{|}$$_{x}$. ■ Proof of Theorem 4. √$_{n}$ ( ˆ F$_{y}$$_{|}$$_{x}$ − F$_{y}$$_{|}$$_{x}$, ψ$_{1}$ ) $_{, ...,}$√$_{n}$ ( ˆ F$_{y}$$_{|}$$_{x}$ − F$_{y}$$_{|}$$_{x}$, ψ$_{l}$ ) 39