LAW OF THE SUM OF BERNOULLI RANDOM VARIABLES

Let Delta(n) be the set of all possible joint distributions of n Bernoulli random variables X(1), ... , X(n). Suppose that Delta(n), which is a simplex in the 2(n)-dimensional space, is endowed with the normalized Lebesgue measure mu(n). Suppose also that the integer n is large. Then we show that there is a subset Delta of Delta(n), whose measure mu(n)(Delta) is very close to 1, such that if the joint distribution of (X(1), ... , X(n)) is in Delta, then the law of the sum X(1) + ... + X(n) is close to the binomial law B( n, 1/2). This result does not need any independence assumption. Next, we show a result of the same kind when Delta(n) is endowed with another probability measure nu(n).