On the Proximity of Sequences of Probability Distributions
Annotatsiya
Many problems in probability theory and mathematical statistics are reduced to the summation of independent random variables. In turn, the latter is closely connected with the study of sequences of distribution functions for sums of independent random variables. It is well known that the distribution function of the sum of independent random variables is a convolution (composition) of the distribution functions of the summands. It is natural to consider sequences of distribution functions $$\left\{F_{n},n\geq 1\right\}$$ and $$\left\{G_{n},n\geq 1\right\}$$ to be close if $$F_{n}-G_{n}\to 0$$ properly or weakly. In the proposed work, is used modifications of the Rotar numerical characteristic, which is introduced by the author of this paper. The problems related to the proximity of sequences of convolutions of distributions $$\{F_{n}\}$$ and $$\{G_{n}\}$$ generated by the corresponding sums of independent random variables forming ’’a triangular array’’ are studied.