By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F _n ,Φ) between the standard normal distribution function Φ and the distribution function F _n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β³. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β³+0.429)≤0.33477(1+0.429)β³<0.4784β³ by virtue of the condition β³≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequality. As corollaries, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.

This paper describes the history of the search for unconditional and conditional upper bounds of the absolute constant in the Berry–Esseen inequality for sums of independent identically distributed random variables. Computational procedures are described. New estimates are presented from which it follows that the absolute constant in the classical Berry–Esseen inequality does not exceed 0.5129.

We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an M_t/M_t/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.

This paper deals with the mathematical basis of the possibility of using a Student distribution in problems of descriptive statistics. We especially separate the case where the Student distribution parameter (``the number of degrees of freedom") is small. We show that the Student distribution with arbitrary ``number of degrees of freedom" can be obtained as the limit when the sample size is random. We emphasize the possibility of using a family of Student distributions as a comfortable model with heavy tails since in this case many relations, in particular, a likelihood function, have the explicit form (unlike stable laws). As an illustration of the possibilities of statistical analysis based on the family of Student distributions, we consider a problem of statistical estimation of the center of the Student distribution under the assumption that the parameter of the form (the number of degrees of freedom) is known. We consider equivariant estimators of the center of the Student distribution based on order statistics, M-estimators, and maximum likelihood estimators, calculate their asymptotic relative efficiency, and study the behavior of the Student distribution when ``the number of degrees of freedom" tends to zero.