Represent ( − 1) d$_{x}$ ∫ ˆ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( ˆ F$_{x}$ ( x ) ) d ( ˆ F$_{x}$ ( x ) ) as ( − 1) d$_{x}$ { ∫ F$_{x,y}$∂ $^{d$_{x}$}$ψ ( F$_{x}$ ) d ( F$_{x}$ ) + ∫ F$_{x,y}$ [( ∂ $^{d$_{x}$}$∂$^{d$_{x}$}$ψ ) ( F$_{x}$ ) ( ˆ F$_{x}$ − F$_{x}$ ) + r ( ˆ F$_{x}$ − F$_{x}$ ) $^{2}$] d ( F$_{x}$ ) + ∫ F$_{x,y}$∂ $^{d$_{x}$}$ψ ( F$_{x}$ ) d ( ˆ F$_{x}$ − F$_{x}$ ) + F$_{x,y}$ ( ∂ $^{d$_{x}$}$∂$^{d$_{x}$}$ψ ) ( ˜ F$_{x}$ ) ( ˆ F$_{x}$ − F$_{x}$ d ( ˆ F$_{x}$ − F$_{x}$ ) + ( ˆ F$_{x,y}$ − F$_{x,y}$ ) ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ) dF$_{x}$ + ∫ ( ˆ F$_{x,y}$ − F$_{x,y}$ ) ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ) d ( ˆ F$_{x}$ − F$_{x}$ ) + ∫ ( ˆ F$_{x,y}$ − F$_{x,y}$ )( ∂ $^{d$_{x}$}$∂$^{d$_{x}$}$ψ ) ( ˜ F$_{x}$ ) ( ˆ F$_{x}$ − F$_{x}$ ) dF$_{x}$ + ( ˆ F$_{x,y}$ − F$_{x,y}$ )( ∂ $^{d$_{x}$}$∂$^{d$_{x}$}$ψ ) ˜ F$_{x}$ ˆ F$_{x}$ − F$_{x}$ d ( ˆ F$_{x}$ − F$_{x}$ ) where ˜ F$_{x}$ represents an intermediate value and takes values in (0 , 1) d$_{x}$ ; by properties of ψ ∈ D ( W ) the function ( ∂ $^{d$_{x}$}$∂$^{d$_{x}$}$ψ ) ( ˜ F$_{x}$ ) is bounded. Then √$_{n}$ ( ˆ F$_{y}$$_{|}$$_{x}$ − F$_{y}$$_{|}$$_{x}$, ψ ) can be expressed as Q$_{ψ}$ ( √$_{n}$$_{(}$ ˆ F$_{x}$ − F$_{x}$ ) $_{,}$√$_{n}$ ( ˆ F$_{xy}$ − F$_{xy}$ )) + n − 1 2 R ( √$_{n}$$_{(}$ ˆ F$_{x}$ − F$_{x}$ ) $_{,}$√$_{n}$ ( ˆ F$_{xy}$ − F$_{xy}$ )) , Q$_{ψ}$ ( √$_{n}$$_{(}$ ˆ F$_{x}$ − F$_{x}$ ) $_{,}$√$_{n}$ ( ˆ F$_{xy}$ − F$_{xy}$ )) = ∫ F$_{x,y}$ [( ∂ $^{d$_{x}$}$ψ ) ( F$_{x}$ )] $_{d}$√$_{n}$ ( ˆ F$_{x}$ − F$_{x}$ ) + ∫ √$_{n}$$_{(}$ ˆ F$_{x,y}$ − F$_{x,y}$ )[( ∂ $^{d$_{x}$}$ψ ) ( F$_{x}$ )] dF$_{x}$ + ∫ F$_{x,y}$ [( ∂ $^{d$_{x}$}$∂$^{d$_{x}$}$ψ ) ( F$_{x}$ $_{)]}$√$_{n}$ ( ˆ F$_{x}$ − F$_{x}$ ) d ( F$_{x}$ ) and R ( ., . ) is a bounded function. Since the limit process of √$_{n}$ ( ˆ F$_{.}$ − F$_{.}$ ) is U$_{.}$, a Brownian bridge, and the where 38