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Let $X, Y$ be continuous random variables with their distributions $F_X, F_Y$, finite second moments and correlation $\rho$.

I would like to find a smallest possible set $S$ (set such that its Lebesgue measure is as small as possible) satisfying $$P(X+Y\in S)\geq 0.9.$$ The issue: $S$ can be ONLY a function of $F_X, F_Y, \rho$.

Would you have some idea on how to do that (possibly under some additional assumptions or at least some nontrivial special cases)?

A first intuitive guess gives me $S_{intuitive}:= S_x(0.95) + S_y(0.95)$, where $S_x(0.95)=(quantile_X(0.025), quantile_X(0.975))$ is a smallest set such that $P(X \in S_x(\alpha)) \geq\alpha$ and $S_y$ is defined analogously. I am using notation $A+B = \{a+b: a\in A, b\in B\}$ for two sets $A,B$. Can you find anything better?

Albert Paradek
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  • What does shortest possible $S$ mean? – Kavi Rama Murthy Dec 02 '24 at 09:24
  • edited it. if its an interval, we just want it to be as short as possible, meaning minimizing its Lebesgue measure – Albert Paradek Dec 02 '24 at 09:34
  • $X$ and $Y$ are not independent, and their correlation and marginal CDFs will not be sufficient to describe their joint distribution or that of $X+Y$. – Henry Dec 02 '24 at 11:24
  • If $X$ and $Y$ are independent, you can often do better, but the lack of independence is an obstacle. So, to find worst case behaviour, you can try $Y = \lambda X$. – D. Thomine Dec 02 '24 at 11:36
  • One way to get better bounds, at least in some cases, might be to distribute the risk in a different way: use $S_X (1-\alpha) + S_Y (1-\beta)$ with $\alpha+\beta = 0.1$, and optimize in $\alpha$. It might be interesting to see what the optimum is for Gaussians with different variances. – D. Thomine Dec 02 '24 at 11:38
  • @D.Thomine if they are independent, can we do better? that is not clear to me as well, can you elaborate? – Albert Paradek Dec 02 '24 at 17:13

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