Method of Moments and Sums of Random Indicators

Using the method of moments, we derive two theorems on the normal approximation of the sum of $n$ random indicators in a scheme of series in which the joint distribution of indicators may change with increasing $n$. The first theorem provides conditions for the convergence of all moments to the moments of the normal distribution as $n\to \infty $, and the second theorem provides accuracy estimates for the normal approximation in the uniform metric. To demonstrate the efficiency of the results, we use the particle allocation problem and the problem on the accuracy of the normal approximation for the number of solutions to random nonlinear inclusions.