Using Correlation :Random Variables as a Normed Space (Banach, Hilbert maybe.?)

[Bacle]

Hi, everyone:

I have been curious for a while about the similarity between the correlation
function and an inner-product: Both take a pair of objects and spit out
a number between -1 and 1, so it seems we could define a notion of orthogonality
in a space of random variables, so that correlation-0 random variables are orthogonal.

Does anyone know how far we can take this analogy, i.e., can we use correlation
as an inner-product to define a norm ( autocorrelation Corr(X,X)), and therefore
a normed space.?

Thanks.

 

[mathman]

The set, of all random variables over a given probability space, which have a finite second moment, is a Hilbert space.

This is a special case of the following: Given a measure space, the set of all square integrable functions is a Hilbert space.

 

[Bacle]

Thanks.
What I was thinking about was more along the lines that
correlation as defined seems to mimic the cosine of the
angle between two vectors, given that -1<= Corr(X,Y)<=1
I wonder if there is some inner-product thatt would give rise to
this, as is the case with, e.g, R^n (n>1). I know we have some restrictions
since the above expression is not linear in neither x nor Y;
still, I wonder if there is a way of making it work.

 

[mathman]

In finite dimensional Euclidean space (a,b)=|a||b|cos(x), where | | denotes length and x is the angle between the vectors. The covariance is equivalent to cos(x).