Unimodality of Ordinary Multinomials and Maximal Probabilities of Convolution Powers of Discrete Uniform Distribution

We establish the unimodality and the asymptotic strong unimodality of the ordinary multinomials and give their smallest mode leading to the expression of the maximal probability of convolution powers of the discrete uniform distribution. We conclude giving the generating functions of the sequence of generalized ordinary multinomials and for an extension of the sequence of maximal probabilities for convolution power of discrete uniform distribution. Since the eighteenth century, the expression of the convolution power of the discrete uniform distribution has been very well known (e.g. de Moivre in 1711, see [11, 3rd ed., 1756] or [12, 1731]). This probability distribution arises in many practical situations including, in particular, games with equal chance, random affectation of tasks for many servers, and random walks. It is well known, see Dharmadhikari & Joak-Dev [7, 1988, p. 108-109.], that the convolution of two discrete unimodal distributions may be non unimodal. However, if these distributions are symmetric, we obtain a symmetric unimodal distribution. It is a discrete analog of Wintner’s Theorem [19, 1938]. Knowing that the convolution power of the discrete uniform distribution is symmetric unimodal, the determination of the maximal probability (mode) of such a distribution and its argument remains a question for consideration. As a recent work on the problem, one can see the article by Mattner & Roos [9, 2007] where they establish the upper bound for the maximal probability cq,L < √ 6/πq (q + 2)L (cq,L being the maximal probability of the L-th convolution power of the discrete uniform distribution on {0, 1, ..., q}). Before them, there were several works aiming at finding such a bound. For example, Siegmund-Schultze & von Weizsäcker [15, 2007] proved the existence of a constant A such that cq,L < A/ (q + 1) √ L and gave an application of these upper bounds in the construction of a polygonal recurrence of a two-dimensional random walk. MSC 2000 Subject Classification. Primary 60C05, 05A10; secondery 11B39, 11B65