Wikipedia states here that the chi distribution $\chi _{k}$ with $k$ degrees of freedom converges to the standard normal distribution
${\displaystyle \lim _{k\to \infty }{\tfrac {\chi _{k}-\mu _{k}}{\sigma _{k}}}{\xrightarrow {d}}\ N(0,1)\,} $
I cannot find any references or equivalent statements on the internet. Is this statement true and, if yes, does someone know a reference or a proof?
Best Answer
The $x\ge0$ PDF $\tfrac{1}{2^{k/2-1}\Gamma(k/2)}x^{k-1}e^{-x^2/2}$ is asymptotic to $e^{-S}$ with$$S:=x^2/2-(k-1)\ln x+(k/2-1)\ln2+\ln\Gamma(k/2).$$As per Laplace's method, for large $k$ we can approximate the log-PDF as a quadratic, based on its first two derivatives. Since $S_x=x-\tfrac{k-1}{x}$ is $0$ at $x=\sqrt{k-1}$, and $S_{xx}=1+\tfrac{k-1}{x^2}>0$, this value of $x$ minimizes $S$ and maximizes the PDF. At this value of $x$,$$S=(k-1)/2-\tfrac12(k-1)\ln(k-1)+(k/2-1)\ln2+\ln\Gamma(k/2)$$and $S_{xx}=2$, so in general$$S\approx (k-1)/2-\tfrac12(k-1)\ln(k-1)+(k/2-1)\ln2+\ln\Gamma(k/2)+\left(x-\sqrt{k-1}\right)^2,$$and we've approximated the $\chi_k$ distribution as $N(\sqrt{k-1},\,\tfrac12)$. This Gaussian approximation can be restated as $\frac{\chi_k-\mu_k}{\sigma_k}$ being approximately Gaussian, with mean $0$ and variance $1$, and hence $\approx N(0,\,1)$.