ОЦЕНКА СКОРОСТИ СХОДИМОСТИ В ЗАКОНЕ БОЛЬШИХ ЧИСЕЛ ДЛЯ ГАММА-РАСПРЕДЕЛЕННЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ

Abstract

The paper considers the problem of estimating the rate of convergence in the law of large numbers for the case when the initial set of random variables is distributed according to the law of the gamma distribution. The problem is urgent due to the fact that with a small number of initial random variables, accurate and close to the true values are the values obtained on the basis of averaging, in particular, if the receipt of each additional value is associated with significant resource costs. The main result of the paper contains estimates for the modulus of difference in distribution function of the mean value for the set of N random variables in the original population, where N is arbitrary, and distribution function of their limiting value, which is a constant (mean value). The result includes three cases: when the argument of distribution function is greater than the average value; when it is equal to it and when it is less than the average value. Estimates are obtained for the modulus of difference of distributions, which depend not only on the number of random variables N, but also on the argument of distribution function. The dependence of the obtained estimate on the argument of distribution function has an exponential character, and on the volume of the set N this dependence makes about the root of N. For convenience of practical application, and also for solving the inverse problem on the basis of the obtained result, estimating the modulus of the difference of distributions is simplified. On the basis of the simplified estimates obtained, the solution of the following inverse problem is given: to find the minimum volume of the string N at which the modulus of the difference of distributions (the accuracy of estimating the mean value on the basis of the mean value) does not exceed a given (small) value. The paper presents a formula for finding the specified minimum volume N, and an algorithm for finding the exact value of N for the estimate under consideration.