where the inequality follows from the fact that e$_{C}$ N − ( m − 1) ≤ 1. We shall show that this limit can be bounded from below by a positive constant that depends on ( x$_{0}$, y$_{0}$ ) and R but not on I . Since a positive lower bound on expression (A.2) can be obtained: The partition I either contains the cell C , or a group of cells that divide C . By Jensen’s inequality, it follows that if the partition I contains a group of cells that divide C , the score is made larger, since for any partition of the cell C , C = ∪$_{l}$ C$_{l}$ , where the first inequality follows since h ( x, y ) − f ( x ) g ( y ) > 0 in A , and the sec- ond inequality follows since the minimum value is c > 0. Therefore, it follows that 1 N − m +1 ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ converges uniformly almost surely to a positive constant greater than c ′ = c $^{2}$R$^{2}$/ 4 , 44 $^{(}$o$_{C}$ − e$_{C}$ e$_{C}$ ) 2 =   ∑ l e$_{C}$$_{l}$ ( o$_{C}$$_{l}$ e$_{C}$$_{l}$ − 1 ) $^{∑}$$_{h}$e$_{C}$$_{h}$   2 ≤ $^{∑}$$_{l}$e$_{C}$$_{l}$ ( o$_{C}$$_{l}$ e$_{C}$$_{l}$ − 1 ) 2 $^{∑}$$_{l}$e$_{C}$$_{l}$ = ∑ l ( o$_{C}$$_{l}$ − e$_{C}$$_{l}$ ) 2 e$_{C}$$_{l}$ e$_{C}$ , Pr$^{(}$ lim N →∞ 1 N − m + 1 ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ ≥ c ′ ) = 1 . (A.4) lim N →∞ [ ∫ { ( x,y ): x ∈ ( x$_{i}$,x$_{j}$ ] ,y ∈ ( y$_{i}$,y$_{j}$ ] } { h ( x, y ) − f ( x ) g ( y ) } dxdy$^{]}$$^{2}$ ≥ [ ∫ { ( x,y ): x ∈ ( x$_{0}$ + R/ 4 ,x$_{0}$ +3 R/ 4] ,y ∈ ( y$_{0}$ + R/ 4 ,y$_{0}$ +3 R/ 4] } { h ( x, y ) − f ( x ) g ( y ) } dxdy$^{]}$$^{2}$ ≥ c 2 ∫ { ( x,y ): x ∈ ( x$_{0}$ + R/ 4 ,x$_{0}$ +3 R/ 4] ,y ∈ ( y$_{0}$ + R/ 4 ,y$_{0}$ +3 R/ 4] } dxdy = c $^{2}$R$^{2}$/ 4 , (A.3) { ( 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$_{i}$, x$_{j}$ ] , y ∈ ( y$_{i}$, y$_{j}$ ] } ,