freedom, it thus follows that for x > D , Let x = c ′ $_{8}$( N − m + 1) − D/ 4. Then for N large enough, x > D . It thus follows that By Bonferroni’s inequality, where the last inequality follows from (A.11). Since ( N m ) is at most O ( N$^{√}$ $^{N}$), and since it follows that the expression in (A.12) goes to zero as N → ∞ . Since we found contingency tables for which under the alternative the test statis- tic T I N − m +1 converges uniformly almost surely to a positive constant greater than c ′ (A.4), it follows that M DDP m × m N − m +1 converges uniformly almost surely to a positive constant greater than c ′ when the null is false. From (A.12) it follows that as N → ∞ , with lim$_{N}$$_{→∞}$ m √ N = 0, the probability that the test statistics M DDP m × m N − m +1 will be above c ′ 2 goes to zero when the null is true. It follows that the null hypothesis will be rejected with asymptotic probability one when it is false. 48 Pr$_{H}$$_{0}$$^{(}$M DDP m × m ≥ c ′ 2 ( N − m + 1)$^{)}$ ≤ ∑ I∈ Π DDP m Pr$_{H}$$_{0}$$^{(}$T I ≥ c ′ 2 ( N − m + 1)$^{)}$ ≤$^{(}$ N m$^{)}$ e − ( c ′ 8 ( N − m +1) − ( m − 1) $^{2}$/ $^{4)}$, (A.12) e − ( c ′ 8 ( N − m +1) − ( m − 1) $^{2}$/ 4) = O ( e − ( c ′ $_{8}$N $^{)}$) , Pr$_{H}$$_{0}$ ( T I ≥ c ′ 2 ( N − m + 1)) ≤ e − ( c ′ 8 ( N − m +1) − D/ 4) ≤ e − ( c ′ 8 ( N − m +1) − ( m − 1) $^{2}$/ 4) (A.11) Pr$_{H}$$_{0}$ ( T I − D ≥ 4 x ) ≤ e − $^{x}$. (A.10)