TITLE: Is the sum of independent unimodal random variables still unimodal?
QUESTION [6 upvotes]: Is the sum of independent unimodal random variables still unimodal? If yes, can you please give me some hint on why this holds? If no, can you show me some counter-example and suggest under what condition the sum remains unimodal? Thank you in advance.

REPLY [6 votes]: In addition to Henning's answer, here is a continuous distribution example of unimodal density such the sum is not unimodal:
$$
   f_X(x) = \frac{1}{182} \max\left( \frac{128}{x^2}, 42 - 5x \right) \mathbf{1}_{x \ge 1}
$$

The density $f_{X+Y}(z)$ is unsightly, so its explicit form is suppressed in the snapshot.