Comments by the authors on Volume 9, article
R26 (November 6, 2002):
After our paper was published, we learned, through James Fill, that
the same Markov chain was studied more than two decades ago by Kingman
in the context of the "common ancestor problem." The upper
bound in Theorem 5 of Kingman  is already better than ours, and in
the intervening years the result has been generalized and extended.
See, for example, Theorem 3 of Donnelly, and Möhle,.
(We gratefully acknowledge Simon Tavaré's help in locating these
references.) There is an enormous literature that can be traced back to
Kingman's work . We do not attempt a review here, but simply acknowledge
our lack of priority.
- P. Donnelly, Weak convergence to a Markov chain with
an entrance boundary: ancestral processes in population genetics,
The Annals of Probability
19 No.3, (1991) 1102-1117.
- P. Donnelly and S. Tavaré, Coalescents and genealogical
structure under neutrality, Annual Review of Genetics
29 (1995) 401-425.
- R.C. Griffiths, Exact sampling distributions from the infinite
neutral alleles model, Advances in Applied Probability
11 (1979) 326-354.
- R.C. Griffiths, Lines of descent in the diffusion approximation
of neutral Wright-Fisher models, Theoretical Population Biology
17 (1980) 37-50.
- J.F.C. Kingman, The Coalescent, Stochastic Proc.
Appl. 13 (1982) 235-248.
- J.F.C. Kingman, On the genealogy of large populations.
Essays in statistical science. J. Appl. Probab. 19A
- J.F.C. Kingman, Exchangeability and the evolution of large
populations, in Exchangeability in probability and statistics,
pp. 97-12, North-Holland, Amsterdam-New York, 1982.
- J.F.C. Kingman, Mathematics of genetic diversity.
CBMS-NSF Regional Conference Series in Applied Mathematics
34. Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, Pa., 1980.
- M. Möhle, The time back to the most recent common ancestor
in exchangeable population models, preprint submitted for publication
(2002). (Currently available on
Martin Möhle's website.)
- M. Möhle, Total variation distances and rates of convergence
for ancestral coalescent processes in exchangeable population
models, Adv. Appl. Prob. 32 (2000) 983-993.
- S. Tavaré, Line-of-descent and genealogical processes and
their applications in population genetics models, Theoretical
Population Biology 26 (1984) 119-164.
- S. Tavaré , Ancestral inference from DNA Sequence data,
Statistical Science 4 No.3 (1994) 307-319.
- G.A. Watterson, On the number of segregating sites in genetic
models without recombination, Theoretical Population Biology
7 (1975) 256-276.
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