Before revision:  In particular, we study tests of randomness (i.e., that trials are i.i.d.) which choose as test statistics (i) the difference between the proportions of successes that directly follow k consecutive successes and k consecutive failures or (ii) the difference between the proportion of successes following k consecutive successes and the proportion of successes .
After revision:  In particular, we study tests of randomness (i.e., that trials are i.i.d.) which choose as test statistics (i) the difference between the proportions of successes that directly follow k consecutive successes and k consecutive failures or (ii) the difference between the proportion of successes or the proportion of successes .