Proving the Property of Covariance Function: (r(n)-r(m))^2≤2r(0)(r(0-r(n-m)))

[trenekas]

Hi all. My task is to prove the property of covariance function:

##(r(n)-r(m))^2≤2r(0)(r(0-r(n-m)))##

My solution:

##1) (r(n)-r(m))^2=r(n)^2-2r(n)r(m)+r(m)^2##
##2) 2r(0)(r(0)-r(n-m)))=2r(0)^2-2r(0)r(n-m)##

From covariance function properties I know that ##2r(0)^2≥r(n)^2+r(m)^2##
So now I just need to prove that ##r(0)r(n-m)≤r(n)r(m)##
But don't know how to do that. Any thoughts? I'm not sure, maybe my way of solution is bad and I need to find other one. Any help would be appreciate.

 

[Ray Vickson]

Hi all. My task is to prove the property of covariance function:

##(r(n)-r(m))^2≤2r(0)(r(0-r(n-m)))##

My solution:

##1) (r(n)-r(m))^2=r(n)^2-2r(n)r(m)+r(m)^2##
##2) 2r(0)(r(0)-r(n-m)))=2r(0)^2-2r(0)r(n-m)##

From covariance function properties I know that ##2r(0)^2≥r(n)^2+r(m)^2##
So now I just need to prove that ##r(0)r(n-m)≤r(n)r(m)##
But don't know how to do that. Any thoughts? I'm not sure, maybe my way of solution is bad and I need to find other one. Any help would be appreciate.

What is the function ##r(n)##; that is, how is it defined, and what is its relation to anything?

 

[trenekas]

What is the function ##r(n)##; that is, how is it defined, and what is its relation to anything?

If I understand you I'll try to answer.
##r(n)## is covariance function of stationary process. ##r(n)## show the covariance between two random variables and ##n## is operator of distance.
##r(n)=EX_nX_0-EX_nEX_0##
Because process X is stationary, then ##EX_nX_0-EX_nEX_0=EX_{n+h}X_h-EX_{n+h}EX_h##
And some properties:
##r(0)≥0##
##r(h)=r(-h)##
##|r(h)|≤r(0)##
But it does not help me to solve the problem :(

 

[Ray Vickson]

If I understand you I'll try to answer.
##r(n)## is covariance function of stationary process. ##r(n)## show the covariance between two random variables and ##n## is operator of distance.
##r(n)=EX_nX_0-EX_nEX_0##
Because process X is stationary, then ##EX_nX_0-EX_nEX_0=EX_{n+h}X_h-EX_{n+h}EX_h##
And some properties:
##r(0)≥0##
##r(h)=r(-h)##
##|r(h)|≤r(0)##
But it does not help me to solve the problem :(

OK, so you have a stationary process in which
[tex] \text{Cov}(X_{n+k},X_n) = r(k), k=0,1,2, \ldots . [/tex]
In that case, the notation in the question still does not make sense: the RHS involves ##r(0-r(n-m))= -r(n-m)##, and I cannot fathom this. If we stick to your definition, this means
[tex] \text{Cov}(X_k,X_{k-(-r(n-m))}),[/tex]
and there is no reason for a time like ##t = k -(-r(n-m)) = k + r(n-m)## to be integer-valued.

Did your problem statement have a typo in it? Did you really mean ##r(0) - r(n-m)?##

 

[trenekas]

Oh my God. I made a mistake :) In the first line it should be ##r(0)*(r(0)−r(n−m))##