Classical and non-classical central limit theorems for random sums of independent random variables of a double sequence

TL;DR

This paper extends classical and non-classical central limit theorems to random sums of independent, non-identically distributed variables in double sequences, with new conditions for their validity, impacting various fields like statistics and finance.

Contribution

It generalizes and extends existing CLTs for random sums of independent variables in double sequences, providing new conditions for their applicability.

Findings

Extended classical CLTs to double sequences of random sums.

Provided new conditions ensuring the validity of these CLTs.

Generalized known results to broader settings.

Abstract

Since the appearance of H. Robbins article (1948), the central limit theorems for random sums have been studied for about 70 years. The central limit theorems for random sums of independent random variables play a very important role in various disciplines such as statistics, financial mathematics, insurance, etc. The purpose of this paper is to randomize some well-known classical and non-classical central limit theorems for sums of independent (not necessarily identically distributed) random variables of a double sequence, with the conditions for determining their validity. The results obtained in this paper are extensions and generalizations of known ones.