×

Asymptotics of the uniform measure on simplices, and random compositions and partitions. (English. Russian original) Zbl 1081.60009

Funct. Anal. Appl. 37, No. 4, 273-280 (2003); translation from Funks. Anal. Prilozh. 37, No. 4, 39-48 (2003).
The limiting behaviour of the uniform distribution generated by the coordinates of the “typical point” on an \(m\)-dimensional simplex is studied as \(m\) goes to infinity. It is shown that the limit is the exponential distribution with parameter one. The related problem of the limiting behavior of uniform measures on the compositions and partitions of positive integers is investigated as well. Consider a composition of a positive integer \(n\) into \(m\) parts. It is shown that the uniform measure on such compositions tends to the exponential distribution if \(n,m\to\infty\), \(m=o(n)\). The same result holds for partitions of positive integers if \(m=o(n^{1/2})\). As a corollary, a refinement of the result of Erdős and Lehner about the asymptotic absence of repeated summands in partitions is given.

MSC:

60D05 Geometric probability and stochastic geometry
60C05 Combinatorial probability
05A17 Combinatorial aspects of partitions of integers
PDFBibTeX XMLCite
Full Text: DOI