Branching Processes and Neutral Evolution

The Galton-Watson branching process has its roots in the problem of extinction of family names which was given a precise formulation by F. Galton as problem 4001 in the Educational Times (17, 1873). In 1875, an attempt to solve this problem was made by H. W. Watson but as it turned out, his conclusion was incorrect. Half a century later, R. A. Fisher made use of the Galton-Watson process to determine the extinction probability of the progeny of a mutant gene. However, it was J. B. S. Haldane who finally gave the first sketch of the correct conclusion. J. B. S. Haldane also predicted that mathematical genetics might some day develop into a "respectable branch of applied mathematics" (quoted in M. Kimura & T. Ohta, Theoretical Aspects of Population Genetics. Princeton, 1971). Since the time of Fisher and Haldane, the two fields of branching processes and mathematical genetics have attained a high degree of sophistication but in different directions. This monograph is a first attempt to apply the current state of knowledge concerning single-type branching processes to a particular area of mathematical genetics: neutral evolution. The reader is assumed to be familiar with some of the concepts of probability theory, but no particular knowledge of branching processes is required. Following the advice of an anonymous referee, I have enlarged my original version of the introduction (Chapter Zero) in order to make it accessible to a larger audience. G6teborg, Sweden, November 1991.

0. Introduction.- 0.1. The general branching process.- 0.2. The neutral theory.- 0.3. The results.- 1. The Construction of the Process.- 1.1. The basic probability space.- 1.2. Processes counted with random characteristics.- 1.3. Special cases.- 1.4. Decompositions.- 1.5. General assumptions.- 2. Labelled Branching Processes.- 2.1. Labelled branching populations.- 2.2. The fate of the ancestral label.- 3. The Number of Distinct Labels.- 3.1. The total number of labels.- 3.2. The number of present labels.- 3.3. A Bellman-Harris application.- 3.4. The lattice case.- 4. The Label Process.- 4.1. The embedded label process.- 4.2. The oldest labels.- 4.3. The most frequent labels.- 5. The Limiting Stable Case.- 5.1. The stable pedigree space.- 5.2. The stable pedigree measure.- 5.3. Sampling among living individuals.- 6. The Process of Mutant Ancestors.- 6.1. The age of the label of an RSI.- 6.2. A renewal process.- 7. A Measure of Relatedness.- 7.1. The mean size of a label.- 7.2. The distribution of the size of a label.- 7.3. The probability of identity by descent.- 8. Infinite Sites Labels.- 8.1. The mean size.- 8.2. Another measure of relatedness.- 9. Related Individuals.- 9.1. Genealogical distances.- 9.2. Distances between relatives.- 10. Embeddings in Branching Processes.- 10.1. Trees and general branching processes.- 10.2. The branching property.- 10.3. Generalized individuals.- 11. Extensions.- 11.1. Other sampling schemes.- 11.2. Further developments.- Appendices.- A. Basic convergence theorems.- B. The lattice case.- A List of Symbols and Conventions.- References.