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freedom, it thus follows that for x > D , 

<img src='content_image/318677.jpg'>

Let x = c ′ $_{8}$( N − m + 1) − D/ 4. Then for N large enough, x > D . It thus follows that 

<img src='content_image/318676.jpg'>

By Bonferroni’s inequality, 

<img src='content_image/318674.jpg'>

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 → ∞ . 

<img src='content_image/318675.jpg'>

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. 

