On nonuniform estimates of accuracy of normal approximation for distributions of some random sums under relaxed moment conditions

Nonuniform estimates are presented for the rate of convergence in the central limit theorem for sums of a random number of independent identically distributed random variables. Two cases are studied in which the summation index (the number of summands in the sum) has the binomial or Poisson distribution. The index is assumed to be independent of the summands. The situation is considered where the information that only the second moments of the summands exist is available. Particular numerical values of the absolute constants are presented explicitly. Also, the sharpening of the absolute constant in the nonuniform estimate of the rate of convergence in the central limit theorem for sums of a nonrandom number of independent identically distributed random variables is announced for the case where the summands possess only second moments.