Assume $X$ and $Y$ are two dependent random variables and we do not have the joint distribution of these two. Is there an upper/lower bound for the PDF of $X+Y$? I found a paper which provides bounds but it needs additional random variable $Z$ with known distribution as a function of $X$ and $Y$.
2026-04-13 17:38:32.1776101912
Bounds for PDF of Sum of Two Dependent Random Variables
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In general there is no lower bound other than $0$, and no upper bound. Worse, $X+Y$ might not even have a PDF, it might be discrete (e.g. try $Y = -X$).