Refined distributional limit theorems for compound sums

TL;DR

This paper systematically presents refined distributional limit theorems for compound sums, distinguishing between process theory and classical summation, using a unified proof approach based on probability rules and auxiliary theorems.

Contribution

It introduces a unified approach to proving distributional limit theorems for compound sums, including refinements, applicable to ergodic semi-Markov systems.

Findings

Unified proof method for limit theorems

Refinements for distributional convergence

Application to semi-Markov systems

Abstract

The paper is a sketch of systematic presentation of distributional limit theorems and their refinements for compound sums. When analyzing, e.g., ergodic semi-Markov systems with discrete or continuous time, this allows us to separate those aspects that lie within the theory of random processes from those that relate to the classical summation theory. All these limit theorems are united by a common approach to their proof, based on the total probability rule, auxiliary multidimensional limit theorems for sums of independent random vectors, and (optionally) modular analysis.