Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T09:29:50.349Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Problems in Mathematical Genealogy

Published online by Cambridge University Press:  05 September 2017

David G. Kendall
Get accessRights & Permissions [Opens in a new window]

Abstract

After a general review of symmetric reversibility for countable-state continuous-time Markov chains the author shows that the birth-death-and-immigration process is symmetrically reversible and further that it remains so even when the description of the present state is refined to include a list of the sizes of all ‘families’ alive at the epoch in question. This result can be useful in genealogy because the operational direction of time there is the negative one. In view of the symmetric reversibility, some of the questions which face the genealogist can be answered without further calculation by quoting known results for the process with the usual (‘forward’ instead of ‘backward’) direction of time.

Further topics discussed include social mobility matrices, surname statistics, and Colin Rogers' ‘problem of the Spruces’.

Information

Type
Part VIII — Probability Models in the Humanities
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975, pp. 325 - 345
Copyright
Copyright © 1975 Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Bartlett, M. S. and Kendall, D. G. (1951) On the use of the characteristic functional in the analysis of some stochastic processes occurring in physics and biology. Proc. Camb. Phil. Soc. 47, 65–76.Google Scholar
[2] Bühler, W. J. (1972) The distribution of generations and other aspects of the family structure of branching processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 3, 463–480.Google Scholar
[3] Edwards, A. W. F. (1970) Estimation of the branch points of a branching diffusion process. J. R. Statist. Soc. B 32, 155–174.Google Scholar
[4] Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecol. 12, 42–58.Google Scholar
[5] Guppy, H. B. (1890) The Homes of Family Names in Great Britain. London.Google Scholar
[6] Hammersley, J. M. (1951) The sums of products of the natural numbers. Proc. London Math. Soc. (3)1, 435–452.Google Scholar
[7] Kendall, D. G. (1948) On some modes of population growth leading to R. A. Fisher's logarithmic series distribution Biometrika 35, 6–15.Google Scholar
[8] Kendall, D. G. (1949) Stochastic processes and population growth. J. R. Statist. Soc. B 11, 230–264 and following discussion.Google Scholar
[9] Kendall, D. G. (1950) Random fluctuations in the age-distribution of a population whose development is controlled by the simple birth and death process. J. R. Statist. Soc. B 12, 278–285.Google Scholar
[10] Kendall, D. G. (1952) Les processus stochastiques de croissance en biologie. Ann. Inst. H. Poincare 13, 43–108.Google Scholar
[11] Kendall, D. G. (1959) Unitary dilations of one-parameter semigroups etc. Proc. London Math. Soc. (3)9, 417–431.Google Scholar
[12] Kendall, D. G. (1971) Markov Methods. (Cambridge lectures, as yet unpublished.) Google Scholar
[13] Kolmogorov, A. N. (1936) Zur Theorie der Markoffschen Ketten. Math. Ann. 112, 155–160.Google Scholar
[14] Mckendrick, A. G. (1914) Studies in the theory of continuous probabilities, with special reference to its bearing on natural phenomena of a progressive nature. Proc. London Math. Soc. (2) 13, 401–416.Google Scholar
[15] Quenouille, M. H. (1949) A relation between the logarithmic, Poisson, and negative binomial series, Biometrics 5, 162–164.Google Scholar
[16] Redmonds, G. (1973) Yorkshire: West Riding (English Surnames Series, 1). Phillimore, London.Google Scholar
[17] Rogers, C. (1973) Unpublished lecture. (See Yorks. Archaeol. Soc. Family History Newsletter 5, 5.)Google Scholar
[18] Berger, J. and Snell, J. L. (1957) On the concept of equal exchange. Behavioral Science 2, 111–118.Google Scholar
[19] Who, J. Recollections and Anticipations. Unpublished MS, n.d., possibly available (uncatalogued) in Bodleian Library, Oxford.Google Scholar
Kendall, D. G. (1975) Branching processes before (and after) 1873: the genealogy of genealogy. Bull. London Math. Soc. To appear. (This contains further comments on mathematical problems in genealogy, surname statistics, etc.) Google Scholar