Is this a known measure of "effective degrees of freedom" in regression? Suppose $x,y$ are I.I.D random variables in $d$ dimensions. Consider the following quantity:
$$R=\frac{(E\|x\|^2)^2}{E\langle x, y\rangle^2}$$
equivalently for empirical distribution with data matrix $X$
$$R=\frac{(\operatorname{Tr}X'X)^2}{\operatorname{Tr}X'XX'X}$$
Does this quantity come up in statistics literature?
This came up recently under ML as "effective rank $R_0$" (Bartlett Lemma 5). It's similar to quantity mentioned in wikipedia under Satterthwaite approximation, except where the hat matrix $H=X(X'X)^{-1}X'$ is replaced with $X$.

This quantity seems to reduce to "Welch-Satterthwaite degrees of freedom" is the limit of variance of $x,y$ going to 0
