Some results on unimodality of distributions (Q915296)

The author proves the following theorem: Let \(p=(p_ n)\) be a unimodal sequence of nonnegative numbers, and let \(G_ 1\) and \(G_ 2\) be nonnegative, nondecreasing real functions. Then the sequence \(g=(g_ n)\) by \[ g=G_ 1(p)*G_ 2(\bar p) \] is a unimodal sequence. Here \(\bar p=(\bar p_ n)=(p_{-n})\), * denotes convolution, and \((G_ 1(p))_ n=G_ 1(p_ n)\). There is an analogous result for densities. The proofs are based on the proof of a lemma saying that \(X-X'\) is unimodal if X and \(X'\) are i.i.d. and unimodal. This fact for continuous distributions can also be found in \textit{S. Dharmadhikari} and \textit{K. J. Joag-dev's} book, ``Unimodality, convexity and applications''. (1988; Zbl 0646.62008). Some of the corollaries can also be found there. The final section is mainly devoted to high convolutions, and simple examples are given of distributions (both discrete and continuous) such that no convolution power is unimodal. The author mentions the work by \textit{P. L. Brockett} and \textit{J. H. B. Kemperman}, Ann. Probab. 10, 270-277 (1982; Zbl 0481.60021), but does not quite do justice to these earlier results. He is unaware of earlier examples by \textit{N. G. Ushakov} [Theory Probab. Appl. 27, 361-362 (1982; Zbl 0505.60021)], which are rather similar to one of his examples. Finally, a counterexample is given to a conjecture by \textit{P. Medgyessy} concerning the unimodality of certain compound Poisson distributions.