Density Function of Weighted Sum of Chi-Square Variables with Trigonometric Weights

TL;DR

This paper derives the exact density function and large-sample approximation for a weighted sum of chi-square variables with trigonometric weights, comparing it to Gaussian distribution.

Contribution

It provides a novel exact and approximate density function for a specific weighted chi-square distribution with trigonometric weights.

Findings

Exact density function derived for the distribution.

Large N approximation of the density function.

Comparison with Gaussian distribution highlights differences.

Abstract

We have investigated a weighted chi-square distribution of the variable $\xi$ which is a weighted sum of squared normally distributed independent variables whose weights are cosines of angles $\phi_k=2\pi k/N$, where $k \in \{0,1,...,N-1\}$ and $N$ is the number of the freedom degrees. We have found the exact expression for the density function of this distribution and its approximation for large $N$. The distribution is compared with the Gaussian distribution.