Let $X_1, \dots, X_n$ be random variables, for some integer $n > 2$. Then, $$\text{Var}\left(\sum_{i=1}^nX_i\right) = \sum_{i=1}^n \text{Var}(X_i) + \sum_{i \neq j} \text{Cov}(X_i, X_j) $$

There is a natural interest in the special case when the $X_i$ are such that $$\sum_{i \neq j} \text{Cov}(X_i, X_j) = 0 \quad \Leftrightarrow \quad \text{Var}\left(\sum_{i=1}^nX_i\right) = \sum_{i=1}^n \text{Var}(X_i)$$

Now, the condition $$\text{Cov}(X_i, X_j) = 0, \,\forall\,i \neq j\tag{1}$$ clearly implies the condition $$\sum_{i\neq j} \text{Cov}(X_i, X_j) = 0 \;.\tag{2}$$

Is the converse true?

Alternatively, can someone give me an example where $(2)$ holds but $(1)$ fails?

(My guess is that $(2)$ does not imply $(1)$, since I realize that, in general, $\sum_1^{n>1} a_i = 0$ does not imply that all the $a_i$ are zero, but I can't readily come up with an example of this general fact when the summands are the terms $\text{Cov}(X_i, X_j),\;(i\neq j)$.)

(My remaining questions are moot if, contrary to what I suspect, $(2)$ does imply, and is therefore equivalent to, $(1)$.)

When the random variables $X_1, \dots, X_n$ are described as being "uncorrelated", does this mean that they satisfy $(1)$, or just that they satisfy $(2)$?

Lastly, I've often seen the term "pairwise uncorrelated", which clearly means that $(1)$ holds, is there a name for the condition in $(2)$ (when $(1)$ is not assumed)?


Consider three random variables with covariance matrix $$\left[\begin{matrix}1 & a & 0\\a & 1 & -a\\ 0 & -a & 1\end{matrix}\right]$$ which has leading principal minors $1$, $1-a^2$, and $1-2a^2$ and thus is positive definite (as all covariance matrices must be) provided that $\vert a\vert < \frac{1}{\sqrt{2}}$. Obviously this satisfies $$\sum_{i\neq j} \text{Cov}(X_i, X_j) = 0 \tag{2}$$ but not $$\text{Cov}(X_i, X_j) = 0, \,\forall\,i \neq j.\tag{1}$$

As noted in my (now-deleted) comment, correlation is a pairwise property and so $n$ random variables are uncorrelated means that $(1)$ holds: every pair of distinct random variables is uncorrelated. As far as I know, there is no name for random variables for which $(2)$ holds but $(1)$ does not. In the example provided, the random variables are not significantly correlated in the sense that $Y$ "fails to explain" more than half the variance of either $X$ or $Z$.

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    $\begingroup$ +1. Generally, such covariance matrices can be derived from matrices $\mathbb{\Sigma}$ having these properties: (1) they are positive semi-definite symmetric Real matrices and (2) the kernel of $\mathbb{\Sigma}^*$ is nontrivial, where $\mathbb{\Sigma}^*$ is obtained by zeroing out the diagonal elements of $\mathbb{\Sigma}$. For correlation matrices like yours, which have unit diagonals, this means $\mathbb{\Sigma}$ has $1$ for an eigenvalue. Sure enough, your matrix--with eigenvalues $1$ and $1\pm a\sqrt{2}$--qualifies. $\endgroup$ – whuber Jul 7 '13 at 21:08

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