Estimation of the number of terms of the normal approximation of the sums of independent random variables

The paper solves the problem of determining the number of independent random variables with the same mathematical expectations and different variances, the sum of which has a normal distribution law with a given accuracy. A similar problem is considered for an arithmetic mean sample from a normal probability distribution. The theorem is proved and the corollary from it is obtained. The proof of the theorem is based on the decomposition of characteristic functions into a Maclaurin series. Based on the dependencies obtained in the theorem, tables are calculated to determine the required number of terms for a given accuracy for different mean square deviations of sample observations. Graphs of the obtained dependencies are constructed. The dependence of the required number of terms on the accuracy is approximated by a polynomial of the sixth degree. The theorem proved in the article and the obtained dependencies can be used in testing, monitoring, observation and diagnostics systems.