Abstract
This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having "large'' eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of the spectrum of the urn's replacement matrix and examples of each case are treated. We study the cases of so-called cyclic urns in any dimension and $m$-ary search trees for $m \geq 27$.
Highlights
We consider Polya-Eggenberger urns with balls of s different types, s being any integer ≥ 2
One inspects the colour of the drawn ball, places it back into the urn and adds other balls following invariably the same rule. This rule is given by the replacement matrix
Theorem 1 leads to a classification of large Polya-Eggenberger urns in five types depending on the form of their asymptotics
Summary
We consider (generalized balanced) Polya-Eggenberger urns with balls of s different types (or colours), s being any integer ≥ 2. A Polya-Eggenberger urn defined by U1 and R being given, one can consider it as a random walk in Rs having U1 as initial point, the increment Un+1 − Un at time n being at random one of R’s rows, the probability of the k-th one to be chosen being equal to Un,k/(|U1| + (n − 1)S) (proportion of balls of colour k after the (n − 1)-st draw) Adopting this point of view, we standardize the process (or the urn) the following way. We classify large Polya-Eggenberger urns with regard to their asymptotics, give some generic example of each case and some other new results about particular families of urns (general twodimensional urn, cyclic urns, m-ary search trees)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
