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February, 1982 On the Unimodality of High Convolutions
Patrick L. Brockett, J. H. B. Kemperman
Ann. Probab. 10(1): 270-277 (February, 1982). DOI: 10.1214/aop/1176993933


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 $n$.


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Patrick L. Brockett. J. H. B. Kemperman. "On the Unimodality of High Convolutions." Ann. Probab. 10 (1) 270 - 277, February, 1982.


Published: February, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0481.60021
MathSciNet: MR637396
Digital Object Identifier: 10.1214/aop/1176993933

Primary: 60E05
Secondary: 60F05

Keywords: sums of i.i.d. random variables , Unimodal distributions

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 1 • February, 1982
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