Colored coalescent theory

Pages: 833 - 845, Issue Special, August 2005

 Abstract        Full Text (150.4K)              

Jianjun Tian - Mathematical Biociences Institute, The Ohio State University, Columbus, OH 43210, United States (email)
Xiao-Song Lin - Department of Mathematics, University of California, Riverside, Riverside, CA 92521, United States (email)

Abstract: We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertices coalesce. Explicit computations of the expectation and the cumulative distribution function of the coalescent time are carried out. For example, when $x=1/2$, for a sample of $n$ colored individuals, the expected time for the colored coalescent process to reach a black MRCA or a white MRCA, respectively, is $3-2/n$. On the other hand, the expected time for the colored coalescent process to reach a MRCA, either black or white, is $2-2/n$, which is the same as that for the standard Kingman coalescent process.

Keywords:  genealogical tree, coalescent theory, colored coalescent theory, cumulative distribution function, random tree.
Mathematics Subject Classification:  Mathematics Subject Classification

Received: September 2004;      Revised: April 2005;      Published: September 2005.