# On convergence of the sample correlation matrices in high-dimensional data

## TL;DR

This paper analyzes the convergence properties of sample correlation matrices in high-dimensional data, revisiting and extending key theorems to better understand their asymptotic behavior.

## Contribution

It revisits and simplifies proofs of existing theorems on correlation matrix convergence and generalizes a theorem for moments and convergence rates.

## Key findings

- Established four main theorems in full generality
- Simplified proofs of key results
- Generalized a theorem for moments and convergence rates

## Abstract

In this paper, we consider an estimation problem concerning the matrix of correlation coefficients in context of high dimensional data settings. In particular, we revisit some results in Li and Rolsalsky [Li, D. and Rolsalsky, A. (2006). Some strong limit theorems for the largest entries of sample correlation matrices, The Annals of Applied Probability, 16, 1, 423-447]. Four of the main theorems of Li and Rolsalsky (2006) are established in their full generalities and we simplify substantially some proofs of the quoted paper. Further, we generalize a theorem which is useful in deriving the existence of the pth moment as well as in studying the convergence rates in law of large numbers.