Almost sure asymptotics for the number of types for simple Xi-coalescents

作者：Freund Fabian^*

来源：Electronic Communications in Probability, 2012, 17: 1-11.

DOI：10.1214/ECP.v17-1704

• almost sure convergence
• coalescent
• external branches
• mutation

Let K-n be the number of types in the sample {1, ... , n} of a Xi-coalescent Pi = (Pi(t))(t %26gt;= 0) with mutation and mutation rate r %26gt; 0. Let Pi((n)) be the restriction of Pi to the sample. It is shown that M-n/n, the fraction of external branches of Pi((n)) which are affected by at least one mutation, converges almost surely and in L-p (p %26gt;= 1) to M := integral(infinity)(0) re(-rt) S(t)dt, where S-t is the fraction of singleton blocks of Pi(t). Since for coalescents without proper frequencies, the effects of mutations on non-external branches is neglectible for the asymptotics of K-n/n, it is shown that K-n/n --%26gt; M for n --%26gt; infinity in L-p (p %26gt;= 1). For simple coalescents, this convergence is shown to hold almost surely. The almost sure results are based on a combination of the Kingman correspondence for random partitions and strong laws of large numbers for weighted i.i.d. or exchangeable random variables.

• 出版日期2012-1-6