What can we say about the ratio of two Minima/Maxima?
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I have been reading my old script about extreme value theory as well as I've been searching the web, but I could not find any question like that; What I am wondering about:
Assume we have given some random variable $X$ with iid realizations $x_1,...,x_n$ and $y_1,...,y_n$. What can we say about:
$limlimits_{n to infty}frac{min(x_1,...,x_n)}{min(y_1,...,y_n)}$
or
$limlimits_{n to infty}frac{max(x_1,...,x_n)}{max(y_1,...,y_n)}$
Of course we are only interested in random variables where it is non-trivial; e.g. if $X$ is upper bounded by some constant $c$, the ratio over the maxima will of course converge to 1..
Does it e.g. converge to 1 for extremely light tailed distributions like the normal distribution? What about heavy-tailed distributions like the log-normal distribution?
probability probability-theory asymptotics extreme-value-theorem
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
$endgroup$
add a comment |
$begingroup$
I have been reading my old script about extreme value theory as well as I've been searching the web, but I could not find any question like that; What I am wondering about:
Assume we have given some random variable $X$ with iid realizations $x_1,...,x_n$ and $y_1,...,y_n$. What can we say about:
$limlimits_{n to infty}frac{min(x_1,...,x_n)}{min(y_1,...,y_n)}$
or
$limlimits_{n to infty}frac{max(x_1,...,x_n)}{max(y_1,...,y_n)}$
Of course we are only interested in random variables where it is non-trivial; e.g. if $X$ is upper bounded by some constant $c$, the ratio over the maxima will of course converge to 1..
Does it e.g. converge to 1 for extremely light tailed distributions like the normal distribution? What about heavy-tailed distributions like the log-normal distribution?
probability probability-theory asymptotics extreme-value-theorem
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
$endgroup$
$begingroup$
Perhaps you want the limit in front of the fraction. Otherwise, for distributions with unbounded support, these simplify to $frac{infty}{infty}$. Although, limit the limit as written is likely the wrong notion, you really want some convergence in distribution.
$endgroup$
– Michael Burr
Dec 18 '18 at 9:35
$begingroup$
RIght, sorry; Corrected it
$endgroup$
– J.Doe
Dec 18 '18 at 9:37
add a comment |
$begingroup$
I have been reading my old script about extreme value theory as well as I've been searching the web, but I could not find any question like that; What I am wondering about:
Assume we have given some random variable $X$ with iid realizations $x_1,...,x_n$ and $y_1,...,y_n$. What can we say about:
$limlimits_{n to infty}frac{min(x_1,...,x_n)}{min(y_1,...,y_n)}$
or
$limlimits_{n to infty}frac{max(x_1,...,x_n)}{max(y_1,...,y_n)}$
Of course we are only interested in random variables where it is non-trivial; e.g. if $X$ is upper bounded by some constant $c$, the ratio over the maxima will of course converge to 1..
Does it e.g. converge to 1 for extremely light tailed distributions like the normal distribution? What about heavy-tailed distributions like the log-normal distribution?
probability probability-theory asymptotics extreme-value-theorem
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
$endgroup$
I have been reading my old script about extreme value theory as well as I've been searching the web, but I could not find any question like that; What I am wondering about:
Assume we have given some random variable $X$ with iid realizations $x_1,...,x_n$ and $y_1,...,y_n$. What can we say about:
$limlimits_{n to infty}frac{min(x_1,...,x_n)}{min(y_1,...,y_n)}$
or
$limlimits_{n to infty}frac{max(x_1,...,x_n)}{max(y_1,...,y_n)}$
Of course we are only interested in random variables where it is non-trivial; e.g. if $X$ is upper bounded by some constant $c$, the ratio over the maxima will of course converge to 1..
Does it e.g. converge to 1 for extremely light tailed distributions like the normal distribution? What about heavy-tailed distributions like the log-normal distribution?
probability probability-theory asymptotics extreme-value-theorem
probability probability-theory asymptotics extreme-value-theorem
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
edited Dec 18 '18 at 9:38
J.Doe
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
asked Dec 18 '18 at 9:29
J.DoeJ.Doe
304110
304110
$begingroup$
Perhaps you want the limit in front of the fraction. Otherwise, for distributions with unbounded support, these simplify to $frac{infty}{infty}$. Although, limit the limit as written is likely the wrong notion, you really want some convergence in distribution.
$endgroup$
– Michael Burr
Dec 18 '18 at 9:35
$begingroup$
RIght, sorry; Corrected it
$endgroup$
– J.Doe
Dec 18 '18 at 9:37
add a comment |
$begingroup$
Perhaps you want the limit in front of the fraction. Otherwise, for distributions with unbounded support, these simplify to $frac{infty}{infty}$. Although, limit the limit as written is likely the wrong notion, you really want some convergence in distribution.
$endgroup$
– Michael Burr
Dec 18 '18 at 9:35
$begingroup$
RIght, sorry; Corrected it
$endgroup$
– J.Doe
Dec 18 '18 at 9:37
$begingroup$
Perhaps you want the limit in front of the fraction. Otherwise, for distributions with unbounded support, these simplify to $frac{infty}{infty}$. Although, limit the limit as written is likely the wrong notion, you really want some convergence in distribution.
$endgroup$
– Michael Burr
Dec 18 '18 at 9:35
$begingroup$
Perhaps you want the limit in front of the fraction. Otherwise, for distributions with unbounded support, these simplify to $frac{infty}{infty}$. Although, limit the limit as written is likely the wrong notion, you really want some convergence in distribution.
$endgroup$
– Michael Burr
Dec 18 '18 at 9:35
$begingroup$
RIght, sorry; Corrected it
$endgroup$
– J.Doe
Dec 18 '18 at 9:37
$begingroup$
RIght, sorry; Corrected it
$endgroup$
– J.Doe
Dec 18 '18 at 9:37
add a comment |
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$begingroup$
Perhaps you want the limit in front of the fraction. Otherwise, for distributions with unbounded support, these simplify to $frac{infty}{infty}$. Although, limit the limit as written is likely the wrong notion, you really want some convergence in distribution.
$endgroup$
– Michael Burr
Dec 18 '18 at 9:35
$begingroup$
RIght, sorry; Corrected it
$endgroup$
– J.Doe
Dec 18 '18 at 9:37