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Let $X\in\mathbb{R}^n$ be a zero-mean Gaussian random vector with covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$. And let $Y = \text{sign}(X)\in\{-1,1\}^n$.

Has anyone studied this object $Y$? Does it have a name, or does it show up in some branch of probability? I'm wondering if there's a characterization of the joint density of $Y$ (i.e., suppose $v\in\{-1,1\}^n$, then what is $\mathbb{P}(Y=v)$) in terms of the covariance matrix $\Sigma$. Thanks!

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You could call $Y$ a generalized multi-variate indicator function that gives values $-1$ or $1$ on the different parts of the support. Note you have to choose which sign to assign to $0$ (however, a single point that is not a point mass doesn't change the values of any integrals).

In general there is no closed form for $P(X = v)$. However you can express it as an integral of the Gaussian density over an orthant. You can estimate such an integral efficiently with sampling.

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  • $\begingroup$ Yes, I see. So, for example for $n=2$, $\mathbb{P}(Y_1=1,Y_2=1) = \mathbf{1}(X_1>0,X_2>0) = \mathbb{P}(X_1>0,X_2>0)$, which is the integral of the joint Gaussian density over the top right orthant. Thanks! $\endgroup$ – paulcon Oct 5 '15 at 0:25

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