I'm interested to see a result where for large degree of freedom $m,$ the chi distribution $\chi_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $m.$ The motivation comes from the fact that for large $m, \chi^2_m \sim \mathcal{N}(m,2m)$ approximately and the difference between the corresponding CDF's approach zero as the d.f. $m \to \infty$, as asked and proved in this question. The problem is: we can't apply some sort of continuous mapping theorem here to pass onto square root. I said "some sort of" because we'd be looking here for a case where $X_m - Y_m \to_d 0 \implies g(X_m) - g(Y_m)\to_{d}0$ with $g$ being a continuous, real-valued function (in our case $X_m:=\chi^2_m, Y_m:=\mathcal{N}(m,2m), g:= \sqrt{}$), and this is **not** the continuous mapping theorem.

Regardless, the wiki page of chi-squared distribution seems to refer to this theorem, proved by Ronald Fisher:

**Approximately,** $\sqrt{2\chi^2_m} \sim \mathcal{N}(\sqrt{2m-1},1) \implies \chi_m \sim \mathcal{N}(\sqrt{m-\frac{1}{2}},\frac{1}{\sqrt{2}}).$

Despite the reference to a book on that wiki page, I'm unable to access it and am also unable to otherwise get a reference to this theorem. Do you happen to know the proof or could you refer me to it please? Thank you!