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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

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  • $\begingroup$ 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 $n$ $\endgroup$ Mar 17 '20 at 19:36
  • $\begingroup$ Interesting problem. Both variables are not zero mean necessarily; I suppose. If they are zero mean, the ratio is Cauchy distributed. $\endgroup$ Mar 19 '20 at 21:47
  • $\begingroup$ @Perspectiva8, I wrote that I am not interested in the trivial case you mentioned. $\endgroup$
    – Karl
    Mar 22 '20 at 5:38
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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

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

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