Asymptotics of summands I: square integrable independent random variables

TL;DR

This series of papers thoroughly explores the asymptotic limit theory for sums of independent, square-integrable random variables, providing detailed, modern proofs suitable for both students and researchers.

Contribution

This paper offers a comprehensive, self-contained exposition of the asymptotic behavior of sums of independent square-integrable variables, incorporating recent tools and innovative theorem enhancements.

Findings

Detailed asymptotic limit theory exposition

Modernized proofs with recent tools

Complete and innovative theorem presentations

Abstract

This paper is part of series on self-contained papers in which a large part, if not the full extent, of the asymptotic limit theory of summands of independent random variables is exposed. Each paper of the series may be taken as review exposition but specially as a complete exposition expect a few exterior resources. For graduate students and for researchers (beginners or advanced), any paper of the series should be considered as a basis for constructing new results. The contents are taken from advanced books but the organization and the proofs use more recent tools, are given in more details and do not systematically follow previous one. Sometimes, theorems are completed and innovated