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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?

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    $\begingroup$ One result about the ratio of two independent Chi-square distributed random variables is: let $X \sim \chi^2(k)$ and $Y \sim \chi^2(j)$, then $\dfrac{X/k}{Y/j}$ is $F(k,j)$-distributed. $\endgroup$ – random_guy Mar 8 '15 at 16:51
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    $\begingroup$ How would the CLT apply to a sequence of ratios of random variables? $\endgroup$ – whuber Mar 8 '15 at 17:27
  • $\begingroup$ What's going to infinity here? $n$? $k$? $j$? Do $k$ and $j$ relate to $n$ somehow? $\endgroup$ – Glen_b Mar 9 '15 at 0:01
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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.

Asymptotic arguments:

  • 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

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