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Suppose that we have two independent random variables $X$ and $Y$. We are interested in the following covariance:

$$\mathbb{C}ov\left[X,\mathbb{I}\left[X+Y>0\right]\right], $$

where $\mathbb{I}[\cdot]$ is an indicator function. Can we conclude that this covariance is positive? This paper shows that if $Y=0$, the result is true (the indicator function is a monotonically increasing function).

Is it also true in this more general case?

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    $\begingroup$ I think where it says "positive" you mean "non-negative"? $\endgroup$
    – joriki
    Aug 9, 2016 at 19:42

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The covariance (assuming it exists) is nonnegative by the FKG inequality, since $x$ and ${\mathbb I}[x + y > 0]$ are nondecreasing on $\mathbb R^2$.

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