Abstract
It has been conjectured, for any discrete density function $\{p_j\}$ on the integers, that there exists an $n_0$ such that the $n$-fold convolution $\{p_j\}^{\ast n}$ is unimodal for all $n \geq n_0$. A similar conjecture has been stated for continuous densities. We present several counterexamples to both of these conjectures. As a positive result, it is shown for a discrete density with a connected 3-point integer support that its $n$-fold convolution is fully unimodal for all sufficiently large $n$.
Citation
Patrick L. Brockett. J. H. B. Kemperman. "On the Unimodality of High Convolutions." Ann. Probab. 10 (1) 270 - 277, February, 1982. https://doi.org/10.1214/aop/1176993933
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