# 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

## 1 Answer

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

• D.A. Darling (1955) - The maximum of sums of stable random variables.
• V.B. Nevzorov (1988) - Maximum of cumulative sums for the Cauchy distribution.

Maybe they give some insights for your case. Check also this related question: https://mathoverflow.net/questions/47487/probability-of-the-maximum-levy-stable-random-variable-in-a-list-being-greater

• these papers are not about the problem for which I would like to find an answer.
– Karl
Mar 22 '20 at 5:39