Extreme values of ratios of normal random variables

Question

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

Answer 1

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