×

On the unimodality of high convolutions of discrete distributions. (English) Zbl 0561.60021

The paper deals with the unimodality questions of distributions concentrated on the integers \({\mathbb{Z}}\). Recall that a discrete probability distribution \(\{p_ j\}_{j\in {\mathbb{Z}}}\) is called unimodal if the sequence \(\{p_{j+1}-p_ j\}_{j\in {\mathbb{Z}}}\) has exactly one change of sign. We say that \(\{p_ j\}_{j\in {\mathbb{Z}}}\) is strongly unimodal if the convolution \(\{p_ j\}*\{q_ j\}\) is unimodal for any unimodal discrete distribution \(\{q_ j\}_{j\in {\mathbb{Z}}}\). The authors, among others, proved:
if \(\{p_ j\}_{j\in {\mathbb{Z}}}\) is supported by \(j\in \{0,1,2,...,d\}\), i.e. \(p_ j=0\) for \(j<0\) and \(j>d\) and if \(p_ 0,p_ 1,p_{d-1},p_ d\) are strictly positive then there exists \(n_ 0\in {\mathbb{N}}\) such that for \(n>n_ 0\) the n-fold convolution \(\{p_ j\}^{*n}\) is strongly unimodal.
Reviewer: Z.Jurek

MSC:

60E05 Probability distributions: general theory
60C05 Combinatorial probability
Full Text: DOI