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I am trying to understand this paper where the variance of a couple of Gaussian distributed variables is defined as follows:

$$x_0 \sim \mathcal{N} (\overline{x}_0, \Sigma_{x,0})$$

$$w_k \sim \mathcal{N} (0, \Sigma_w)$$

What do the $\Sigma_{x,0}$ and $\Sigma_w$ actually mean here? I know that they're supposed to refer to the variance because they're within the Normal Distribution density symbol, but what do they actually mean on their own, especially when there is the subscript $x,0$?


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I suspect the subscript was actually $x_0$. The matrix $\Sigma_{x_0}$ would be the variance of a random vector taking values in $\mathbb R^{n\times1}$, and one might have $n=1$ so that it's a real-valued random variable. The variance can be defined as $$ \Sigma_{x_0} = \mathbb E((x_0-\bar x_0) (x_0-\bar x_0)^T), $$ where $\bar x_0 = \mathbb E x_0$ is the expected value. So $\Sigma_{x_0}$ is a nonnegative-definite $n\times n$ matrix. It would appear in the density function, thus: $$ f(x) = \frac{1}{\sqrt{2\pi}^n} (\det\Sigma_{x_0})^{-n/2} \exp\left( \frac{-1}{2} (x-\bar x_0)^T \Sigma_{x_0}(x-\bar x_0) \right). $$ In Feller's famous two-volume book on probability, this matrix is called the variance, since it's the natural generalization to higher dimensions of the scalar-valued variance of scalar-valued random variables. But often it's called the "covariance matrix" or just the "covariance" because its entries are covariances between scalar-valued random variables.

I dislike the use of the notation $\bar x_0$ for the expected value because that's often used for the sample mean, so it's confusing.

Later edit: Now that you've linked to the paper, the answers become apparent. A sequence is defined by recursion: $$ x_{k+1} = Ax_k + \text{some other terms}. $$ Generally if $x\in\mathbb R^n$ is a random vector with variance $\operatorname{var}(x)=\Sigma\in\mathbb R^{n\times n}$ and $A$ is a constant (i.e. non-random) matrix, then $$ \operatorname{var}(Ax) = A\Sigma A^T. $$ That is being used here. And the paper states in $(8)$ that $\Sigma_x$ is the variance of $x$ and $\Sigma_{x,k}$ is the variance of $x_k$, and in $(7)$ that the variance of $x_0$ is $\Sigma_{x,0}$. So you have to be attentive to things like that.

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They use $\Sigma_{x,0}$ all over the place in the paper, not sure what they're trying to imply then. So, in a bit more layman's terms to make sure I got it, $\Sigma_{x}$ would be the matrix of (scalar) variances $\sigma^2$ of each scalar element in the vector? – jbx Feb 27 '13 at 17:54
The diagonal entries are variances. The off-diagonal entries are covariances, and need not be positive, nor even non-negative. – Michael Hardy Feb 28 '13 at 4:55
Another thing that might possibly be meant by $\Sigma_{x,0}$ is the variance of a random vector $x$ under a null hypothesis. But I'd certainly need to know more of the context to say that that's what it is. – Michael Hardy Feb 28 '13 at 4:57
Not sure if you have access to it, its on the first page of this PDF…. – jbx Feb 28 '13 at 11:57

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