Limit Theorems for Random Sums of Random Summands

TL;DR

This paper establishes limit theorems for sums of random variables conditioned on their summands, covering dependent, heavy-tailed, and sampled cases using concentration of measure techniques.

Contribution

It introduces new limit theorems for random sums with dependence and heavy tails, extending classical results like Hoeffding's CLT to more complex settings.

Findings

Proved limit theorems for dependent and heavy-tailed summands.

Extended Hoeffding's combinatorial CLT to new scenarios.

Applied concentration of measure techniques to these results.

Abstract

We prove limit theorems for sums of randomly chosen random variables conditioned on the summands. We consider several versions of the corner growth setting, including specific cases of dependence amongst the summands and summands with heavy tails. We also prove a version of Hoeffding's combinatorial central limit theorem and results for summands taken uniformly from a random sample. These results are proved with concentration of measure techniques.