S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
It is currently Fri, 25 Jul 2014 23:17:10 UTC

All times are UTC [ DST ]




Post new topic Reply to topic  [ 6 posts ] 
Author Message
PostPosted: Wed, 18 Apr 2012 19:05:21 UTC 
Offline
Member

Joined: Sat, 26 Mar 2011 16:45:36 UTC
Posts: 21
How can the following statement be proved?

For any X_1, X_2, \dots sequence of random variables, there exists a deteministic c_1, c_, \dots sequence of numbers such that \frac{X_n}{c_n}\to 0 almost surely.

I would be grateful if you could help me.


Top
 Profile  
 
PostPosted: Wed, 18 Apr 2012 21:11:03 UTC 
Offline
Moderator
User avatar

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 13970
Location: Austin, TX
mrgrieco wrote:
How can the following statement be proved?

For any X_1, X_2, \dots sequence of random variables, there exists a deteministic c_1, c_, \dots sequence of numbers such that \frac{X_n}{c_n}\to 0 almost surely.

I would be grateful if you could help me.


I think this is just Borel-Cantelli, but it's been a while since I did this exercise.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination


Top
 Profile  
 
PostPosted: Thu, 19 Apr 2012 19:36:35 UTC 
Offline
Member

Joined: Sat, 26 Mar 2011 16:45:36 UTC
Posts: 21
Shadow wrote:
mrgrieco wrote:
How can the following statement be proved?

For any X_1, X_2, \dots sequence of random variables, there exists a deteministic c_1, c_, \dots sequence of numbers such that \frac{X_n}{c_n}\to 0 almost surely.

I would be grateful if you could help me.


I think this is just Borel-Cantelli, but it's been a while since I did this exercise.


You might be right, but to tell the truth I don't really see how the lemma could be used in this case. Which form of the lemma do you mean and how does it give the result?

Thank you very much in advance!


Top
 Profile  
 
PostPosted: Thu, 19 Apr 2012 22:14:49 UTC 
Offline
Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 1 Jul 2011 01:17:26 UTC
Posts: 454
mrgrieco wrote:
Shadow wrote:
mrgrieco wrote:
How can the following statement be proved?

For any X_1, X_2, \dots sequence of random variables, there exists a deteministic c_1, c_, \dots sequence of numbers such that \frac{X_n}{c_n}\to 0 almost surely.

I would be grateful if you could help me.


I think this is just Borel-Cantelli, but it's been a while since I did this exercise.


You might be right, but to tell the truth I don't really see how the lemma could be used in this case. Which form of the lemma do you mean and how does it give the result?

Thank you very much in advance!

You haven't put any condition on c_n. To use Borel-Cantelli, define c_n so that ∑P(X_n/c_n) converges.


Top
 Profile  
 
PostPosted: Fri, 20 Apr 2012 06:38:53 UTC 
Offline
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6777
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
mathematic wrote:
You haven't put any condition on c_n. To use Borel-Cantelli, define c_n so that ∑P(X_n/c_n) converges.


X_n/c_n is a random variable and \mathbb{P} only takes event as input. I think you mean something like \displaystyle\sum_n\mathbb{P}\left(\left\lvert\frac{X_n}{c_n}\right\rvert>2^{-n}\right) converges.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


Top
 Profile  
 
PostPosted: Sat, 21 Apr 2012 01:23:05 UTC 
Offline
Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 1 Jul 2011 01:17:26 UTC
Posts: 454
outermeasure wrote:
mathematic wrote:
You haven't put any condition on c_n. To use Borel-Cantelli, define c_n so that ∑P(X_n/c_n) converges.


X_n/c_n is a random variable and \mathbb{P} only takes event as input. I think you mean something like \displaystyle\sum_n\mathbb{P}\left(\left\lvert\frac{X_n}{c_n}\right\rvert>2^{-n}\right) converges.

You are right.


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 6 posts ] 

All times are UTC [ DST ]


Who is online

Users browsing this forum: No registered users


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum

Search for:
Jump to:  
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA