On a central limit theorem for shrunken weakly dependent random variables

TL;DR

This paper establishes a central limit theorem for certain weakly dependent stationary sequences of random variables after applying shrinking operators, extending previous results from i.i.d. cases to dependent sequences.

Contribution

It generalizes the CLT for i.i.d. variables to weakly dependent stationary sequences under mixing conditions with shrinking operators.

Findings

CLT proven for weakly dependent sequences with shrinking operators

Extends previous i.i.d. results to dependent sequences

Applicable under specific mixing conditions

Abstract

A central limit theorem is proved for some strictly stationary sequences of random variables that satisfy certain mixing conditions and are subjected to the "shrinking operators" $U_r(x):=[\max\{|x|-r,0\}]\cdot x/|x|,\ r \ge 0$. For independent, identically distributed random variables, this result was proved earlier by Housworth and Shao.