# Extreme values of ratios of normal random variables

the question is:

Given are two independent sequences of iid normal random variables $$X_i$$ and $$Y_i$$.

Form the ratios $$Z_i=X_i/Y_i$$.

What is known about the extreme value distribution of the $$Z_i$$'s, i.e. $$\max(Z_1,\ldots,Z_n)$$ ? (exclude the trivial case that all have standard normal distributions).

I am looking for a literature reference, since I think somebody must have studied this problem already.

Many thanks!

Karl

• small comment that you can normalize $X_\mu$, $Y_\mu$, and $Y_\sigma$ by dividing by $X_\sigma$ and the problem reduces to four parameters ... those plus $n$ Mar 17 '20 at 19:36
• Interesting problem. Both variables are not zero mean necessarily; I suppose. If they are zero mean, the ratio is Cauchy distributed. Mar 19 '20 at 21:47
• @Perspectiva8, I wrote that I am not interested in the trivial case you mentioned.
– Karl
Mar 22 '20 at 5:38

This is just a comment. I have seen some limiting results for $$\max\left(\sum_{i=1}^n Z_i\right)$$: