The fraction of observed counts in the cell C is a linear combination of empirical cumulative distribution functions and the expected fraction under the null is a function of the marginal cumulative distributions where ˆ F denotes the empirical distribution function based on N − ( m − 1) 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 43 lim N →∞ 1 N − ( m − 1) ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ = lim N →∞ ( o$_{C}$ N − ( m − 1) − e$_{C}$ N − ( m − 1) ) 2 e$_{C}$ N − ( m − 1) ≥ lim N →∞ ( o$_{C}$ N − ( m − 1) − e$_{C}$ N − ( m − 1)$^{)}$$^{2}$ = lim N →∞ [ ∫ { ( x,y ): x ∈ ( x$_{i}$,x$_{j}$ ] ,y ∈ ( y$_{i}$,y$_{j}$ ] } { h ( x, y ) − f ( x ) g ( y ) } dxdy$^{]}$$^{2}$ , (A.2) lim N →∞ ( o$_{C}$ N − ( m − 1) −$^{∫}$ { ( x,y ): x ∈ ( x$_{i}$,x$_{j}$ ] ,y ∈ ( y$_{i}$,y$_{j}$ ] } h ( x, y ) dxdy$^{)}$ = 0 , lim N →∞ { e$_{C}$ N − ( m − 1) −$^{(}$$^{∫}$ { x : x ∈ ( x$_{i}$,x$_{j}$ ] } f ( x ) dx$^{) (}$$^{∫}$ { y : y ∈ ( y$_{i}$,y$_{j}$ ] } g ( y ) dy$^{)}}$ = 0 . (A.1) e$_{C}$ N − ( m − 1) = { ˆ F$_{X}$ ( x$_{j}$ ) − ˆ F$_{X}$ ( x$_{i}$ ) }{ ˆ F$_{Y}$ ( y$_{j}$ ) − ˆ F$_{Y}$ ( y$_{i}$ ) } . o$_{C}$ N − ( m − 1) = ˆ F$_{XY}$ ( x$_{i}$, y$_{i}$ ) + ˆ F$_{XY}$ ( x$_{j}$, y$_{j}$ ) − ˆ F$_{XY}$ ( x$_{i}$, y$_{j}$ ) − ˆ F$_{XY}$ ( x$_{j}$, y$_{i}$ ) ,