ABSTRACT
The performance of various tests and experiments is an inevitable condition of the development of science and applications. Before implementation of a new sort of corn or potato in modern agriculture, one must perform a huge volume of experimental confirmations to verify the advanced properties of the new type. The Local Limit Theorem proved to establish another limit relation of probability theory, the Integral Limit Theorem. Analogous problems exist in pedagogy, medicine, economy, social studies. Everything new must be carefully tested before its wide implementation. One may consider meteorology where a number of rainy days, air temperature and humidity, and other factors for different geographical areas are observed and collected for further statistical inference. In scientific and applied activity one often must perform multiple trials under similar conditions. As a rule, outcomes of previous trials have no influence on the forthcoming outcomes.
