Reconciling two definitions of 'uncorrelatedness'

Question

In this paper, the authors defined uncorrelatedness in the following way:

Let $\mathbf{X}=(X_1,...,X_n)$, and $\mathbf{Y}=(Y_1,...,Y_n)$, where $X_i\sim X$ and $Y_i\sim Y$.

$\mathbf{X},\mathbf{Y}$ are uncorrelated iff $(\mathbf{X}-\bar{X}\mathbf{1})'(\mathbf{Y}-\bar{Y}\mathbf{1})=0$, where $\bar{X}=1/n \sum X_i$ and $\mathbf{1}=(1,...,1)$.

My question is what is the link between this definition of uncorrelatedness and the one usually used (I think), that of (for vectors r.v.) covariance matrix - a matrix where each entry $(i,j)$ is $\text{Corr}(X_i,X_j)$ - being an identity matrix?

Any help would be appreciated.

Answer 1

The authors in the specific paper do not treat the case of statistical correlation and non-correlation, which is only defined in terms of expected values. They are interested in vectors of realizations of two random variables, and the conditions they define are with respect to specific vectors, not the r.v.'s in general. And they explicitly say so in the Introduction of the paper.

Their interest lies in providing geometric intuition and conceptual separation about the three concepts of linear dependence, orthogonality and uncorrelatedness, in a linear algebra context, not a statistical one. So their definitions are suited to this view of these concepts, rather than the statistical ones.

Note how much more strict is this condition for uncorrelatedness of vectors of numbers, compared to the conditions for statistical uncorrelatedness: the later requires only that the condition $(\mathbf{X}-\bar{X}\mathbf{1})'(\mathbf{Y}-\bar{Y}\mathbf{1})=0$ holds "on average"