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Suppose that $(X_1,\dotsc,X_K)^T \sim \mathcal{N}(0, \Sigma)$, with $\mathrm{cov}(X_i, X_j) > 0$ for all $i,j$. Prove that $$ \mathbb{E}[X_K 1\{ X_1> 0, \dotsc, X_{K-1}> 0 \} ] > 0$$ where $1\{ A \}$ is the indicator function of $A$.

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    $\begingroup$ Did you not like "$X_i$" for some reason? $\endgroup$ – Michael May 20 at 20:31
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    $\begingroup$ I observe that if $X_1, ..., X_k$ are random variables then $\{X_1>0, X_2>0, ..., X_{k-1}>0\}$ is an event and so $E[X_K|X_1>0, X_2>0, ..., X_{k-1}>0]$ is not a random variable, it is just an expectation, so it does not make sense to speak of it "holding almost surely." On the other hand $E[X_k|X_1, ..., X_{k-1}]$ is a random variable so it may make sense to talk about some event related to this holding almost surely. $\endgroup$ – Michael May 20 at 20:34
  • $\begingroup$ Apologies, removed almost surely. The statement is now what I intended. $\endgroup$ – Oxonon May 21 at 16:36
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    $\begingroup$ @GabrielRomon Expectations conditioned on events are quite common (eg: math.stackexchange.com/questions/3101793/… math.stackexchange.com/questions/258385/… math.stackexchange.com/questions/248407/…) $\endgroup$ – leonbloy May 21 at 18:00
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    $\begingroup$ Minor observation: Proving $E[X_k|(X_1+...+X_{k-1})>0]$ is easier since (assuming the variables are jointly Gaussian) we can just write $Y=X_1+...+X_{k-1}$ and it reduces to proving the statement on a system of 2 jointly Gaussian variables $(Y, X_k)$. In the spirit of this observation, I wonder if the original statement can be proven by considering $E[X_k|X_1,...,X_{k-1}]$ for (almost) all cases when $X_i\geq 0$ for $i \in \{1, ..., k-1\}$ and then integrating these out. I also wonder if this is the approach that Oxonon alludes to. $\endgroup$ – Michael May 22 at 0:03
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This follows immediately from the more general result in

Pitt, Loren D. "Positively correlated normal variables are associated." The Annals of Probability (1982): 496-499.

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