Convergence of the dispersion of normal distribution
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Let ${X_i}$ be a sequence of independent and identically distributed random variables, with distribution $N(mu,sigma^2)$. Define the random variables
$$
Y_n=frac{max{X_1,dots,X_n}-min{X_1,dots,X_n}}{n}.
$$
Does ${Y_n}$ converge? If yes, to what? If not, is there a power of $n$ (at the denominator) for which it converges to a non-zero number almost certainly? By numerical experimentation, it seems that with $n$ it converges to $0$, but with $log n$ it stays apart from $0$.
I was wondering whether one can recover $sigma$ with this limit process.
It's equivalent (by simmetry) to finding the limit of
$$frac{2cdotmax{X_1,dots,Xn}}{n}$$
which is easier to compute.
probability convergence normal-distribution
asked Nov 22 at 19:18
User
483514
add a comment |
up vote
1
down vote
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Let ${X_i}$ be a sequence of independent and identically distributed random variables, with distribution $N(mu,sigma^2)$. Define the random variables
$$
Y_n=frac{max{X_1,dots,X_n}-min{X_1,dots,X_n}}{n}.
$$
Does ${Y_n}$ converge? If yes, to what? If not, is there a power of $n$ (at the denominator) for which it converges to a non-zero number almost certainly? By numerical experimentation, it seems that with $n$ it converges to $0$, but with $log n$ it stays apart from $0$.
I was wondering whether one can recover $sigma$ with this limit process.
It's equivalent (by simmetry) to finding the limit of
$$frac{2cdotmax{X_1,dots,Xn}}{n}$$
which is easier to compute.
probability convergence normal-distribution
asked Nov 22 at 19:18
User
483514
1
stats.stackexchange.com/questions/105745/… Refer to this question
– John_Wick
Nov 22 at 20:20
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let ${X_i}$ be a sequence of independent and identically distributed random variables, with distribution $N(mu,sigma^2)$. Define the random variables
$$
Y_n=frac{max{X_1,dots,X_n}-min{X_1,dots,X_n}}{n}.
$$
Does ${Y_n}$ converge? If yes, to what? If not, is there a power of $n$ (at the denominator) for which it converges to a non-zero number almost certainly? By numerical experimentation, it seems that with $n$ it converges to $0$, but with $log n$ it stays apart from $0$.
I was wondering whether one can recover $sigma$ with this limit process.
It's equivalent (by simmetry) to finding the limit of
$$frac{2cdotmax{X_1,dots,Xn}}{n}$$
which is easier to compute.
probability convergence normal-distribution
asked Nov 22 at 19:18
User
483514
Let ${X_i}$ be a sequence of independent and identically distributed random variables, with distribution $N(mu,sigma^2)$. Define the random variables
$$
Y_n=frac{max{X_1,dots,X_n}-min{X_1,dots,X_n}}{n}.
$$
Does ${Y_n}$ converge? If yes, to what? If not, is there a power of $n$ (at the denominator) for which it converges to a non-zero number almost certainly? By numerical experimentation, it seems that with $n$ it converges to $0$, but with $log n$ it stays apart from $0$.
I was wondering whether one can recover $sigma$ with this limit process.
It's equivalent (by simmetry) to finding the limit of
$$frac{2cdotmax{X_1,dots,Xn}}{n}$$
which is easier to compute.
probability convergence normal-distribution
probability convergence normal-distribution
asked Nov 22 at 19:18
User
483514
asked Nov 22 at 19:18
User
483514
edited Nov 22 at 20:11
asked Nov 22 at 19:18
User
483514
asked Nov 22 at 19:18
User
483514
asked Nov 22 at 19:18
User
483514
483514
1
stats.stackexchange.com/questions/105745/… Refer to this question
– John_Wick
Nov 22 at 20:20
add a comment |
1
stats.stackexchange.com/questions/105745/… Refer to this question
– John_Wick
Nov 22 at 20:20
1
1
stats.stackexchange.com/questions/105745/… Refer to this question
– John_Wick
Nov 22 at 20:20
stats.stackexchange.com/questions/105745/… Refer to this question
– John_Wick
Nov 22 at 20:20
add a comment |
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stats.stackexchange.com/questions/105745/… Refer to this question
– John_Wick
Nov 22 at 20:20