distributions where ˆ F denotes the empirical cumulative distribution function based on N sample points. By the Glivenko-Cantelli theorem, uniformly almost surely, Therefore, by Slutsky’s theorem and the continuous mapping theorem, we have that uniformly almost surely where the inequality follows from the fact that e$_{C}$ N ≤ 1. We shall show that the limit (A.14) can be bounded from below by a positive constant that depends on ( x$_{0}$, y$_{0}$ ) and R but not on I . Since 50 { ( x, y ) : x ∈ ( x$_{0}$ + R/ 4 , x$_{0}$ + 3 R/ 4] , y ∈ ( y$_{0}$ + R/ 4 , y$_{0}$ + 3 R/ 4] } ⊆ { ( x, y ) : x ∈ ( x ∗ $_{1}$, x ∗ $_{2}$] , y ∈ ( y ∗ $_{1}$, y ∗ $_{2}$] } , lim N →∞ 1 N ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ = lim N →∞ ( o$_{C}$ N − e$_{C}$ N ) 2 e$_{C}$ N ≥ lim N →∞ ( o$_{C}$ N − e$_{C}$ N ) 2 = lim N →∞ [ ∫ { ( x,y ): x ∈ ( x ∗ $_{1}$,x ∗ $_{2}$] ,y ∈ ( y ∗ $_{1}$,y ∗ $_{2}$] } { h ( x, y ) − f ( x ) g ( y ) } dxdy$^{]}$$^{2}$ , (A.14) lim N →∞ $^{(}$o$_{C}$ N −$^{∫}$ { ( x,y ): x ∈ ( x ∗ $_{1}$,x ∗ $_{2}$] ,y ∈ ( y ∗ $_{1}$,y ∗ $_{2}$] } h ( x, y ) dxdy$^{)}$ = 0 , lim N →∞ $^{{}$e$_{C}$ N −$^{(}$$^{∫}$ { x : x ∈ ( x ∗ $_{1}$,x ∗ $_{2}$] } f ( x ) dx$^{) (}$$^{∫}$ { y : y ∈ ( y ∗ $_{1}$,y ∗ $_{2}$] } g ( y ) dy$^{)}}$ = 0 . (A.13) e$_{C}$ N = { ˆ F$_{X}$ ( x ∗ $_{2}$) − ˆ F$_{X}$ ( x ∗ $_{1}$) }{ ˆ F$_{Y}$ ( y ∗ $_{2}$) − ˆ F$_{Y}$ ( y ∗ $_{1}$) } ,