# Asymptotic Distribution of Centralized $r$ When Sampling from Cauchy

## TL;DR

This paper investigates the asymptotic distribution of the centralized empirical correlation coefficient when sampling from Cauchy distributions, providing new theoretical insights into its behavior as sample size grows large.

## Contribution

It derives novel results on the large-sample distribution of the centralized correlation coefficient for Cauchy-distributed variables, a problem previously lacking detailed analysis.

## Key findings

- Derived the asymptotic distribution of the correlation coefficient
- Provided analytical results for large sample sizes
- Enhanced understanding of correlation behavior with Cauchy data

## Abstract

Assume that $X$ and $Y$ are independent random variables, each having a Cauchy distribution with a known median. Taking a random independent sample of size $n$ of each $X$ and $Y$, one can then compute their centralized empirical correlation coefficient $r$. Analytically investigating the sampling distribution of this $r$ appears possible only in the large $n$ limit; this is what we have done in this article, deriving several new and interesting results.