Proof of Convergence in Distribution for random variables with infinite variance
We are asked to prove that given ${X_n}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true:
$$
frac{X_1+X_2 + dots +X_n}{sqrt{nlog n}} xrightarrow{mathcal{D}}N(0,1).
$$
My idea is to use the taylor expansion of the characteristic function. But no matter what I do, I run into trouble with infinity and I cannot prove the convergence of the limit of c.f.
Can anybody give a hint? Thanks so much!
probability probability-distributions convergence central-limit-theorem characteristic-functions
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
add a comment |
We are asked to prove that given ${X_n}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true:
$$
frac{X_1+X_2 + dots +X_n}{sqrt{nlog n}} xrightarrow{mathcal{D}}N(0,1).
$$
My idea is to use the taylor expansion of the characteristic function. But no matter what I do, I run into trouble with infinity and I cannot prove the convergence of the limit of c.f.
Can anybody give a hint? Thanks so much!
probability probability-distributions convergence central-limit-theorem characteristic-functions
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
Later posted on Cross Validated.
– StubbornAtom
Nov 16 '18 at 19:10
add a comment |
We are asked to prove that given ${X_n}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true:
$$
frac{X_1+X_2 + dots +X_n}{sqrt{nlog n}} xrightarrow{mathcal{D}}N(0,1).
$$
My idea is to use the taylor expansion of the characteristic function. But no matter what I do, I run into trouble with infinity and I cannot prove the convergence of the limit of c.f.
Can anybody give a hint? Thanks so much!
probability probability-distributions convergence central-limit-theorem characteristic-functions
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
We are asked to prove that given ${X_n}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true:
$$
frac{X_1+X_2 + dots +X_n}{sqrt{nlog n}} xrightarrow{mathcal{D}}N(0,1).
$$
My idea is to use the taylor expansion of the characteristic function. But no matter what I do, I run into trouble with infinity and I cannot prove the convergence of the limit of c.f.
Can anybody give a hint? Thanks so much!
probability probability-distributions convergence central-limit-theorem characteristic-functions
probability probability-distributions convergence central-limit-theorem characteristic-functions
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
edited Nov 16 '18 at 18:38
David Leigh
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
asked Nov 11 '18 at 20:27
David LeighDavid Leigh
556
556
Later posted on Cross Validated.
– StubbornAtom
Nov 16 '18 at 19:10
add a comment |
Later posted on Cross Validated.
– StubbornAtom
Nov 16 '18 at 19:10
Later posted on Cross Validated.
– StubbornAtom
Nov 16 '18 at 19:10
Later posted on Cross Validated.
– StubbornAtom
Nov 16 '18 at 19:10
add a comment |
1 Answer
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This question was asked an other time on Crosss Validated. One of the ideas of proof is to use Lindeberg central limit theorem to the array of random variables
$$
Y_{n,k}:=X_kmathbf 1left{leftlvert X_krightrvertleqslant nright}
$$
and show that the contribution of the partial sums of $X_kmathbf 1left{leftlvert X_krightrvertgt nright} $ is negligible.
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
This question was asked an other time on Crosss Validated. One of the ideas of proof is to use Lindeberg central limit theorem to the array of random variables
$$
Y_{n,k}:=X_kmathbf 1left{leftlvert X_krightrvertleqslant nright}
$$
and show that the contribution of the partial sums of $X_kmathbf 1left{leftlvert X_krightrvertgt nright} $ is negligible.
add a comment |
This question was asked an other time on Crosss Validated. One of the ideas of proof is to use Lindeberg central limit theorem to the array of random variables
$$
Y_{n,k}:=X_kmathbf 1left{leftlvert X_krightrvertleqslant nright}
$$
and show that the contribution of the partial sums of $X_kmathbf 1left{leftlvert X_krightrvertgt nright} $ is negligible.
add a comment |
This question was asked an other time on Crosss Validated. One of the ideas of proof is to use Lindeberg central limit theorem to the array of random variables
$$
Y_{n,k}:=X_kmathbf 1left{leftlvert X_krightrvertleqslant nright}
$$
and show that the contribution of the partial sums of $X_kmathbf 1left{leftlvert X_krightrvertgt nright} $ is negligible.
This question was asked an other time on Crosss Validated. One of the ideas of proof is to use Lindeberg central limit theorem to the array of random variables
$$
Y_{n,k}:=X_kmathbf 1left{leftlvert X_krightrvertleqslant nright}
$$
and show that the contribution of the partial sums of $X_kmathbf 1left{leftlvert X_krightrvertgt nright} $ is negligible.
answered Jan 4 at 14:50
community wiki
Davide Giraudo
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Later posted on Cross Validated.
– StubbornAtom
Nov 16 '18 at 19:10