Nature of Distribution of a Random Vector whose finite dimensional distributions and distribution of inner...
up vote
1
down vote
favorite
I apologize if this question is vague or trivial. I have a random unit vector $mathbf u$ in $mathbb R^n$ and the following facts about it are true when $nto infty$:
- $u_i to mathcal N(0,1)$ in distribution as $ntoinfty$
for all $i$ - For any finite $k,$ $u_{i_1},u_{i_2}..ldots u_{i_k} to mathcal N(0,mathbf I_k)$ in distribution as $ntoinfty$
- For any sequence of unit deterministic vectors $mathbf a_n,$ $mathbf a^T mathbf u to mathcal N(0,1)$ in distibution as $ntoinfty$
Now I am having a hard time saying anything about the distribution of $mathbf u$ as a whole. Is it close some how to $mathcal N(0,mathbf I_n)$ for a large enough $n$? Can we it converges to the latter in distribution. Is it possible to say something about the difference $|mathbf v - mathbf z|_2$ where $mathbf z$ is some realization of the standard gaussian vector?
probability-theory weak-convergence
asked Nov 14 at 15:22
Arun
1275
add a comment |
up vote
1
down vote
favorite
I apologize if this question is vague or trivial. I have a random unit vector $mathbf u$ in $mathbb R^n$ and the following facts about it are true when $nto infty$:
- $u_i to mathcal N(0,1)$ in distribution as $ntoinfty$
for all $i$ - For any finite $k,$ $u_{i_1},u_{i_2}..ldots u_{i_k} to mathcal N(0,mathbf I_k)$ in distribution as $ntoinfty$
- For any sequence of unit deterministic vectors $mathbf a_n,$ $mathbf a^T mathbf u to mathcal N(0,1)$ in distibution as $ntoinfty$
Now I am having a hard time saying anything about the distribution of $mathbf u$ as a whole. Is it close some how to $mathcal N(0,mathbf I_n)$ for a large enough $n$? Can we it converges to the latter in distribution. Is it possible to say something about the difference $|mathbf v - mathbf z|_2$ where $mathbf z$ is some realization of the standard gaussian vector?
probability-theory weak-convergence
asked Nov 14 at 15:22
Arun
1275
Could you be more precise about your "unit deterministic vector" clause? What tends to infinity there?
– kimchi lover
Nov 14 at 15:42
@kimchilover Sorry, in each case n goes to infinity. To be more precise a is a sequence of unit vectors ...
– Arun
Nov 14 at 16:42
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I apologize if this question is vague or trivial. I have a random unit vector $mathbf u$ in $mathbb R^n$ and the following facts about it are true when $nto infty$:
- $u_i to mathcal N(0,1)$ in distribution as $ntoinfty$
for all $i$ - For any finite $k,$ $u_{i_1},u_{i_2}..ldots u_{i_k} to mathcal N(0,mathbf I_k)$ in distribution as $ntoinfty$
- For any sequence of unit deterministic vectors $mathbf a_n,$ $mathbf a^T mathbf u to mathcal N(0,1)$ in distibution as $ntoinfty$
Now I am having a hard time saying anything about the distribution of $mathbf u$ as a whole. Is it close some how to $mathcal N(0,mathbf I_n)$ for a large enough $n$? Can we it converges to the latter in distribution. Is it possible to say something about the difference $|mathbf v - mathbf z|_2$ where $mathbf z$ is some realization of the standard gaussian vector?
probability-theory weak-convergence
asked Nov 14 at 15:22
Arun
1275
I apologize if this question is vague or trivial. I have a random unit vector $mathbf u$ in $mathbb R^n$ and the following facts about it are true when $nto infty$:
- $u_i to mathcal N(0,1)$ in distribution as $ntoinfty$
for all $i$ - For any finite $k,$ $u_{i_1},u_{i_2}..ldots u_{i_k} to mathcal N(0,mathbf I_k)$ in distribution as $ntoinfty$
- For any sequence of unit deterministic vectors $mathbf a_n,$ $mathbf a^T mathbf u to mathcal N(0,1)$ in distibution as $ntoinfty$
Now I am having a hard time saying anything about the distribution of $mathbf u$ as a whole. Is it close some how to $mathcal N(0,mathbf I_n)$ for a large enough $n$? Can we it converges to the latter in distribution. Is it possible to say something about the difference $|mathbf v - mathbf z|_2$ where $mathbf z$ is some realization of the standard gaussian vector?
probability-theory weak-convergence
probability-theory weak-convergence
asked Nov 14 at 15:22
Arun
1275
asked Nov 14 at 15:22
Arun
1275
edited Nov 17 at 18:08
asked Nov 14 at 15:22
Arun
1275
asked Nov 14 at 15:22
Arun
1275
asked Nov 14 at 15:22
Arun
1275
1275
Could you be more precise about your "unit deterministic vector" clause? What tends to infinity there?
– kimchi lover
Nov 14 at 15:42
@kimchilover Sorry, in each case n goes to infinity. To be more precise a is a sequence of unit vectors ...
– Arun
Nov 14 at 16:42
add a comment |
Could you be more precise about your "unit deterministic vector" clause? What tends to infinity there?
– kimchi lover
Nov 14 at 15:42
@kimchilover Sorry, in each case n goes to infinity. To be more precise a is a sequence of unit vectors ...
– Arun
Nov 14 at 16:42
Could you be more precise about your "unit deterministic vector" clause? What tends to infinity there?
– kimchi lover
Nov 14 at 15:42
Could you be more precise about your "unit deterministic vector" clause? What tends to infinity there?
– kimchi lover
Nov 14 at 15:42
@kimchilover Sorry, in each case n goes to infinity. To be more precise a is a sequence of unit vectors ...
– Arun
Nov 14 at 16:42
@kimchilover Sorry, in each case n goes to infinity. To be more precise a is a sequence of unit vectors ...
– Arun
Nov 14 at 16:42
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2998389%2fnature-of-distribution-of-a-random-vector-whose-finite-dimensional-distributions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Could you be more precise about your "unit deterministic vector" clause? What tends to infinity there?
– kimchi lover
Nov 14 at 15:42
@kimchilover Sorry, in each case n goes to infinity. To be more precise a is a sequence of unit vectors ...
– Arun
Nov 14 at 16:42