Since N$_{1}$/N converges almost surely to f$_{1}$ , and similarly N$_{2}$/N converges almost surely to f$_{2}$ , then for m ≥ 3 finite it follows that ( N 1 m − $_{2}$) N$_{2}$ ( N m − $_{1}$) converges almost surely to a positive constant. Therefore, | Γ | / ( N − 1 m − 1 ) is bounded away from zero. Second, we shall consider the case that m → ∞ . The complement of Γ, Γ $^{C}$, is the set of contingency tables with no points in A$_{1}$ or in A$_{2}$ . An upper bound for | Γ $^{C}$| is Since N$_{1}$/N converges almost surely to f$_{1}$ ∈ (0 , 1), it follows that ( N − N 1 m − ) ( N m − $_{1}$) converges almost surely to zero as m → ∞ . Similarly, since N$_{2}$/N converges almost surely to f$_{2}$ ∈ (0 , 1), it follows that ( N − N 2 m − 1 ) ( N m − $_{1}$) converges almost surely to zero. Therefore, | Γ $^{C}$| / ( N m − 1 ) converges to zero as N → ∞ . Since lim$_{N}$$_{→∞}$ | Γ | / ( N m − 1 $^{)}$is bounded below by a positive constant, it follows from (A.6) that almost surely for some constant c ′′ > 0. Consider now a random permutation ( πy$_{1}$ ) , . . . , ( πy$_{N}$ ) of the y -values y$_{1}$, . . . , y$_{N}$ . Let S π m × m be the test statistic that is computed from the data ( x$_{1}$, πy$_{1}$ ) , . . . , ( x$_{N}$, πy$_{N}$ ). Note that in order to show that lim$_{N}$$_{→∞}$ | Γ | / ( N m − 1 ) is bounded below by a positive constant, since | Γ | = ( N m − 1 $^{)}$−  | Γ $^{C}$| , it is enough to show that | Γ $^{C}$| / ( N m − 1 ) converges to zero as N → ∞ . Simple algebraic manipulations lead to the following expression for ( N − N 1 m − ) ( N m − $_{1}$) : ( 1 − N$_{1}$ N ) · · ·$^{(}$ 1 − N$_{1}$ N − ( m − 2)$^{)}$ ≤$^{(}$ 1 − N$_{1}$ N ) m . $^{(}$N − N$_{1}$ m − 1 ) +$^{(}$ N − N$_{2}$ m − 1 ) . lim N →∞ S$_{m}$$_{×}$$_{m}$ ( N m − 1 $^{)}$( N − m + 1) ≥ c $^{′′}$, (A.7) 46