If the limiting distribution of $X_n \sim \chi^2(k)$ and $Y_n \sim \chi^2(j)$ would their ratio $X_n/Y_n$ also converge to a chi-squared limiting distribution, or would it follow the normal because of CLT?
With finite d.f.
the ratio of two independent chi-squared variates has a beta-prime distribution (also sometimes called a 'beta distribution of the second kind').
if you divide each of the chi-square variates by its df the ratio has an F-distribution.
if $j\to\infty$, you can apply Slutsky's theorem to argue that the F-ratio should go to a $\chi^2_k/k$
if $k$ also $\to\infty$ you can in turn (with appropriate standardization) make an argument that invokes CLT