On the conditions of convergence of the functional from the sum of independent random variables to the functional from the Wiener process

The article finds conditions for convergence of the functional from the sum sequence of series of independent, equally distributed in each series of random variables with values from N - dimensional Euclidean space, to the functional from the Wiener process. The results obtained are further applied to investigate the maximum of partial Erdos-Renyi sums of independent, identically distributed random variables. Sufficient conditions are obtained for the convergence of the distribution function to the limit law.