Chi Distribution

Let $x_n=\vert g_n\vert$ where $g_n$ follows the standard normal distribution $N(0,1)$. Then $x_n$ are Chi-distributed (Section 17.1.3)Let $h(x)=x^2$. Then, for ${\rm Re}(\lambda)<\frac{1}{2}$,

$$\begin{array}{rcl} e(u,\lambda)&=&{\displaystyle \int_u^\infty} ... ... && \;\;\; \; \frac{1}{\sqrt{1-2\lambda}} \mbox{ for } \; u<0. \end{array}$$

Also,

$$\begin{array}{rcl} c(u,\lambda) &=& \frac{1-2\Phi(-u\sqrt{1-2\l... ...-2\lambda}} \;\;\; \mbox{ for } \; u>0; \; 0 \mbox{ for } \; u<0. \end{array}$$

We compare the above results with the SPA in Section 8.2.