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

Abstract

We have investigated a weighted chi-square distribution of the variable ξ which is a weighted sum of squared normally distributed independent variables whose weights are cosines of angles ${{\varphi }_{k}} = {{2\pi k} \mathord{\left/ {\vphantom {{2\pi k} N}} \right. \kern-0em} N},$ where $k \in \left\{ {0,1,...,N - 1} \right\}$ 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.

APPENDIX A

Let us consider the product

$$\prod\limits_{k = 0}^{N - 1} {(\Lambda - } {{\lambda }_{k}}) = {{2}^{N}}\prod\limits_{k = 0}^{N - 1} {(\cos \varphi - } \cos {{\varphi }_{k}}) = {{2}^{{2N}}}\prod\limits_{k = 0}^{N - 1} {\sin \left( {\frac{{{{\varphi }_{k}} + \varphi }}{2}} \right)} \cdot \prod\limits_{k = 0}^{N - 1} {\sin \left( {\frac{{{{\varphi }_{k}} - \varphi }}{2}} \right)} ,$$

(A.1)

where weights are defined by the expression (2) and a new variable $\Lambda = 2\cos \varphi$ is defined.

Using the well-known relation

$${{2}^{{N - 1}}}\prod\limits_{k = 0}^{N - 1} {\sin \left( {x - \frac{{\pi k}}{N}} \right)} = \sin Nx$$

(A.2)

we obtain the following expression for the product (A.1):

$$\prod\limits_{k = 0}^{N - 1} {(\Lambda - } \,\,{{\lambda }_{k}}) = - 4{{\sin }^{2}}\frac{{N\varphi }}{2}.$$

(A.3)

Extracting from (A.3) the factors with nondegenerate weights $\Lambda - {{\lambda }_{0}}$ and $\Lambda - {{\lambda }_{{n + 1}}}$, where ${{\lambda }_{0}} = 2$ and ${{\lambda }_{{n + 1}}} = - 2$, and considering that other weights are twice degenerated, we obtain for the quantity $R = \prod\nolimits_{m = 1}^n {(\Lambda - {{\Lambda }_{m}})}$:

$$R = \frac{1}{{\sqrt {{{\Lambda }^{2}} - 4} }}\prod\limits_{k = 0}^{N - 1} {{{{\sqrt {\Lambda - \lambda } }}_{k}}} = \frac{2}{{\sqrt {4 - {{\Lambda }^{2}}} }}\sin \frac{{N\varphi }}{2}.$$

(A.4)

From (A4) the expression (8) follows

$$\prod\limits_{r \ne m}^n {({{\Lambda }_{m}} - {{\Lambda }_{r}})} = \mathop {\lim }\limits_{\Lambda \to {{\Lambda }_{m}}} \frac{R}{{\Lambda - {{\Lambda }_{m}}}} = \frac{{{{{( - 1)}}^{m}}N}}{{4 - \Lambda _{m}^{2}}}.$$

(A.5)