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AlanB
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Posts: 662
Joined: July 14th, 2002, 3:00 am

Simple Problem?

February 1st, 2012, 3:27 pm

Given a random variable X~N(0,1). What is the correlation coefficient between X and Y=X^2. The simple solution: the square of a Gaussian process is itself a Gaussian process. In this case Y has zero mean and unit variance Y~N(0,1) and thus the correlation coefficient between X and Y is 1. Is there a way of seeing this using Ito's lemma, as follows: dX = dW, dY = 2XdX+(dX)^2. Clearly, <dX>=<dW>=0, <(dX)^2>=dt, <dY>=2<X dX> + dt and <dX dY> = 2<X (dX)^2> + O(dt^1.5) ~ 2<X (dX)^2>. How does one prove that 2<X (dX)^2> = dt?
Last edited by AlanB on January 31st, 2012, 11:00 pm, edited 1 time in total.
 
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croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

Simple Problem?

February 1st, 2012, 3:35 pm

The answer is 0. About your argument, I am able to find a Gaussian process with jumps whose square is not Gaussian, so you should add to your reasoning the qualifier "in the absence of jumps, a Gaussian process...". Hope this helps.
 
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AlanB
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Posts: 662
Joined: July 14th, 2002, 3:00 am

Simple Problem?

February 1st, 2012, 4:33 pm

Interesting as my first gut reaction was that the correlation is zero. The person who posed the problem insists that the correlation is 1. This is, in part, why I'm looking to see if the correlation can be deduced using the Ito lemma analysis I noted in my initial text.QuoteOriginally posted by: crootThe answer is 0. About your argument, I am able to find a Gaussian process with jumps whose square is not Gaussian, so you should add to your reasoning the qualifier "in the absence of jumps, a Gaussian process...". Hope this helps.
Last edited by AlanB on January 31st, 2012, 11:00 pm, edited 1 time in total.
 
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list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

Simple Problem?

February 1st, 2012, 6:35 pm

original problem relates to random variables not to random processes., ie correlation between X and X^2 is equal to E X ^3 = 0
 
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AlanB
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Posts: 662
Joined: July 14th, 2002, 3:00 am

Simple Problem?

February 1st, 2012, 7:46 pm

Forgive my ignorance - "original problem relates to random variables not to random processes" - could you elaborate?I do agree, though: correlation = E X ^3 = 0QuoteOriginally posted by: listoriginal problem relates to random variables not to random processes., ie correlation between X and X^2 is equal to E X ^3 = 0
 
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list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

Simple Problem?

February 1st, 2012, 11:42 pm

QuoteOriginally posted by: AlanBForgive my ignorance - "original problem relates to random variables not to random processes" - could you elaborate?I do agree, though: correlation = E X ^3 = 0QuoteOriginally posted by: listoriginal problem relates to random variables not to random processes., ie correlation between X and X^2 is equal to E X ^3 = 0It is possible that I incorrectly interpreted the problem but if you write X~N(0,1) it means that you deal with Gaussian random variable with parameters 0 and 1 and you looking the correlation between X and X^2. Then you talk about G processes that does not correspond to stated problem. As far as the problem formulated for random variables and not for random processes then answer was presented in terms of random variables.If we deal with random processes then for example it is easy to see that for t < sE w ( t ) w^2 ( s ) = E [ w ( t ) - w ( s ) ] w^2 ( s ) + E w^3 ( s ) = E [ w ( t ) - w ( s ) ] E w^2 ( s ) + E w^3 ( s ) = 0
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