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).

 

[blue_raver22]

I find this idea intriguing and worth exploring further. The use of correlation as an inner-product in defining a norm and a normed space is an interesting concept that has potential applications in statistical analysis and modeling. While the idea of using correlation as an inner-product has been discussed in the literature, there is still much to be explored and understood in terms of its applicability and limitations.

One possible avenue to explore would be to consider the properties of a normed space such as completeness, convergence, and continuity in the context of correlation as an inner-product. This could provide insights into the behavior of random variables and their relationships in a normed space. Additionally, investigating the potential connections between correlation and other mathematical concepts such as Banach or Hilbert spaces could further enhance our understanding of this analogy.

However, it is important to note that while correlation and inner-products share similar properties, they are fundamentally different concepts. Therefore, caution must be exercised when applying this analogy and further research is needed to fully understand its implications and limitations. Nevertheless, the potential for using correlation as an inner-product in defining a normed space is an exciting prospect that warrants further investigation.