Louchard, Guy and Prodinger, Helmut - A combinatorial and probabilistic study of initial and end heights of descents in samples of geometrically distributed random variables and in permutations

dmtcs:386 - Discrete Mathematics & Theoretical Computer Science, January 1, 2007, Vol. 9 no. 1
A combinatorial and probabilistic study of initial and end heights of descents in samples of geometrically distributed random variables and in permutations

Authors: Louchard, Guy and Prodinger, Helmut

In words, generated by independent geometrically distributed random variables, we study the lth descent, which is, roughly speaking, the lth occurrence of a neighbouring pair ab with a>b. The value a is called the initial height, and b the end height. We study these two random variables (and some similar ones) by combinatorial and probabilistic tools. We find in all instances a generating function Ψ(v,u), where the coefficient of vjui refers to the jth descent (ascent), and i to the initial (end) height. From this, various conclusions can be drawn, in particular expected values. In the probabilistic part, a Markov chain model is used, which allows to get explicit expressions for the heights of the second descent. In principle, one could go further, but the complexity of the results forbids it. This is extended to permutations of a large number of elements. Methods from q-analysis are used to simplify the expressions. This is the reason that we confine ourselves to the geometric distribution only. For general discrete distributions, no such tools are available.


Source : oai:HAL:hal-00966508v1
Volume: Vol. 9 no. 1
Section: Analysis of Algorithms
Published on: January 1, 2007
Submitted on: March 26, 2015
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


Share

Browsing statistics

This page has been seen 38 times.
This article's PDF has been downloaded 126 times.