On the Rotar central limit theorem for sums of a random number of independent random variables

TL;DR

This paper extends the Rotar central limit theorem to sums of a random number of independent, not necessarily identically distributed variables, providing conditions for its validity and analyzing the approximation order.

Contribution

It introduces the Rotar CLT for random sums of independent variables and establishes the conditions under which it holds, broadening its applicability.

Findings

The Rotar CLT applies to sums with a random number of summands.

Conditions for the validity of the Rotar CLT are established.

The order of approximation in the Rotar CLT is analyzed.

Abstract

The Rotar central limit theorem is a remarkable theorem in the non-classical version since it does not use the condition of asymptotic infinitesimality for the independent individual summands, unlike the theorems named Lindeberg's and Lindeberg-Feller's in the classical version. The Rotar central limit theorem generalizes the classical Lindeberg-Feller central limit theorem since the Rotar condition is weaker than Lindeberg's. The main aim of this paper is to introduce the Rotar central limit theorem for sums of a random number of independent (not necessarily identically distributed) random variables and the conditions for its validity. The order of approximation in this theorem is also considered in this paper.