B Proof of Theorem 2.4 We want to show that for all ϵ > 0, lim$_{N}$$_{→∞}$ Pr ( ∣ ∣ S ADP m × $_{m}$( L ) N | Π | − I$_{XY}$ ∣ ∣ > ϵ ) = 0 if lim$_{N}$$_{→∞}$ m √ N = 0 and lim$_{N}$$_{→∞}$ m = ∞ , where | Π | = (( N − 1 m − 1 )) 2 is the number of parti- tions. For continuous marginals, the copula function of the joint distribution of ( X, Y ) is unique, denote it by c ( u, v ). The mutual information is the negative copula entropy, H$_{UV}$ = − ∫ c ( u, v ) log c ( u, v ) dudv , Consider an arbitrary fixed partition I = { ( i$_{1}$, j$_{1}$ ) , . . . , ( i$_{m}$$_{−}$$_{1}$, j$_{m}$$_{−}$$_{1}$ ) } ⊂ { 1 . 5 , . . . , N − 0 . 5 } $^{2}$. Recall that C ( I ) is the set of m × m cells that are defined by the partition. For a cell C , let r$_{l}$ ( C ) and r$_{h}$ ( C ) be, respectively, the lowest and highest x -grid integer values in C . Similarly, let s$_{l}$ ( C ) and s$_{h}$ ( C ) be, respectively, the lowest and highest y -grid integer values in C . The entropy of the partition I is The corresponding empirical (plug in) estimator is 54 I$_{XY}$ = ∫ c ( F$_{X}$ ( x ) , F$_{Y}$ ( y )) f ( x ) g ( y ) log c ( F$_{X}$ ( x ) , F$_{Y}$ ( y )) dxdy = ∫ c ( u, v ) log c ( u, v ) dudv = − H$_{UV}$ . (B.1) H I UV = − ∑ C ∈C ( I ) Pr$^{(}$ r$_{l}$ ( C ) N ≤ U ≤ r$_{h}$ ( C ) N , s$_{l}$ ( C ) N ≤ V ≤ s$_{h}$ ( C ) N ) log$^{{}$ Pr$^{(}$ r$_{l}$ ( C ) N ≤ U ≤ r$_{h}$ ( C ) N , s$_{l}$ ( C ) N ≤ V ≤ s$_{h}$ ( C ) N )} . ˆ H I UV = − ∑ C ∈C ( I ) o$_{C}$ N log ( o$_{C}$ N $^{)}$, o$_{C}$ = N ∑ i =1 I ( r$_{l}$ ( C ) ≤ r$_{i}$ ≤ r$_{h}$ ( C ) , s$_{l}$ ( C ) ≤ s$_{i}$ ≤ s$_{h}$ ( C )) .