What does it mean that random variables of mean zero form a Hilbert space with the inner product being the covariance

probability theory

I see that the covariance can be interpreted as a measure of similarity like the dot product of two vectors, and that the correlation coefficient is akin to the cosine in the dot product, while the variance is the vector norm.

Indeed,

  1. Symmetry: $\text{cov}(X, Y)=\text{cov}(Y, X)$
  2. Bilinearity: $\text{cov}(X, aY + bZ)=a\,\text{cov}(X, Y) + b \,\text{cov}(X, Z)$
  3. Positive definitiveness: $ \text{cov}(X, X)=\text{var}(X)\geq 0$…

and it must only zero when the random variable is "zero."

But I am confused about the "mean zero" class equivalence part.

Best Answer

The mean zero requirement is to ensure the positive definiteness, i.e. if $X$ is in this space and $\operatorname{Cov}(X,X)=0$ then $X$ is the zero random variable. (To have a unique "zero random variable" in this sense, you must first take the quotient by $X \sim Y$ if $X=Y$ a.s., which is the usual procedure used in defining the $L^p$ spaces.) If you bundle in all the random variables with finite variance, you don't have this property anymore because all constants will have $\operatorname{Cov}(c,c)=0$.

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