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Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$ out of the $p$ random variables which are mutually independent (i.e theire joint distribution funciton factors as a product of their individual distribution funcitons). Of course, every singleton $\{X_i\}$ forms an independent collection. Thus $1 \le rank(X_1,\ldots,X_p) \le p$, and the upper bound is attained if $X_1,\ldots,X_p$ are independent to begin with.

Question. Is the above notion of rank defined above studied in the literature ? Are there any interesting properties of quantity ? Are the any known proxies for this concept ?

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    $\begingroup$ I don't think you can say much from the covariance matrix. Consider a) $X$ and $Y$ independent uniform variables on $[-1,1]$, or b) $X$ as before, $Y = 2|X|$. They have the same covariance matrix but a) has rank 2 and b) has rank 1. $\endgroup$
    – user44143
    Commented Apr 27, 2020 at 19:16
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    $\begingroup$ Yes, indeed covarance matrices don't capture nonlinear dependence patterns. Removed my comment above cov matrix. $\endgroup$
    – dohmatob
    Commented Apr 27, 2020 at 19:57
  • $\begingroup$ Recall that for $k\ge 3$, pairwise independence is weaker than full ("mutual") independence. What is your motivation for using "pairwise" in your definition? $\endgroup$
    – Jeppe Stig Nielsen
    Commented Apr 27, 2020 at 21:14
  • $\begingroup$ It was a problem with terminology. Removed the "pairwise" adjective. $\endgroup$
    – dohmatob
    Commented Apr 27, 2020 at 23:10

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