The Theory of Branching Processes

It was about ninety years ago that GALTON and WATSON, in treating the problem of the extinction of family names, showed how probability theory could be applied to study the effects of chance on the development of families or populations. They formulated a mathematical model, which was neglected for many years after their original work, but was studied again in isolated papers in the twenties and thirties of this century. During the past fifteen or twenty years, the model and its general­ izations have been treated extensively, for their mathematical interest and as a theoretical basis for studies of populations of such objects as genes, neutrons, or cosmic rays. The generalizations of the GaIton­ Wa,tson model to be studied in this book can appropriately be called branching processes; the term has become common since its use in a more restricted sense in a paper by KOLMOGOROV and DMITRIEV in 1947 (see Chapter II). We may think of a branching process as a mathematical representation of the development of a population whose members reproduce and die, subject to laws of chance. The objects may be of different types, depending on their age, energy, position, or other factors. However, they must not interfere with one another. This assump­ tion, which unifies the mathematical theory, seems justified for some populations of physical particles such as neutrons or cosmic rays, but only under very restricted circumstances for biological populations.

1.- 8.1. Convergence of the sequence {Zn/mn}.- 8.2. The distribution of W.- 8.3· Asymptotic form of P(Zn = 0).- 8.4. Local limit theorems when w > l.- 8.5· Examples.- 9. Asymptotic results when m < 1.- 10. Asymptotic results when m = 1.- 10.1. Form of the iterates basic result for generating functions.- 4. First and second moments; basic assumption.- 5. Positivity properties.- 6. Transience of the nonzero states.- 7. Extinction probability.- 8. A numerical example.- 9. Asymptotic results for large n.- 9.1. Results when ? < 1.- 9.2. The case ? = 1.- 9.3. Results when ? >1.- 10. Processes that are not positively regular.- 10.1. The total number of objects of various types.- 11. An example from genetics.- 12. Remarks.- 12.1. Martingales.- 12.2. The expectation process.- 12.3. Fractional linear generating functions.- III. The general branching process.- 1. Introduction.- 2. Point-distributions and set functions.- 2.1. Set functions.- 3. Probabilities for point-distributions.- 3.1. Rational intervals, basic sets, cylinder sets.- 3.2. Definition of a probability measure on the point-distributions.- 4. Random integrals.- 5. Moment-generating functionals.- 5.1. Properties of the MGF of a random point-distribution.- 5.2. Alternative formulation.- 6. Definition of the general branching process.- 6.1. Definition of the transition function.- 6.2. Notation.- 7. Recurrence relation for the moment-generating functionals.- 8. Examples.- 8.1. The nucleon cascade and related processes.- 8.2. A one-dimensional neutron model.- 9. First moments.- 9.1. Expectations of random integrals.- 9.2. First moment of Zn.- 10. Existence of eigenfunctions for M.- 10.1. Eigenfunctions and eigenvalues.- 11. Transience of Zn.- 12. The case ? ? 1.- 12.1. Limit theorems when ? ? 1.- 13. Second moments.- 13.1. Expectations of random double integrals.- 13.2. Recurrence relation for the second moments.- 13.3. Asymptotic form of the second moment when ? > 1.- 13.4. Second-order product densities.- 14. Convergence of Zn/?n when ? > 1.- 15. Determination of the l.- 16. Another kind of limit theorem.- 17. Processes with a continuous time parameter.- Appendix 1.- Appendix 2.- Appendix 3.- IV. Neutron branching processes (one-group theory, isotropic case).- 1. Introduction.- 2. Physical description.- 3. Mathematical formulation of the process.- 3.1. Transformation probabilities.- 3.2. The collision density.- 3.3. Definition of the branching process.- 4. The first moment.- 5. Criticality.- 6. Fluctuations; probability of extinction; total number in the critical case.- 6.1. Numerical example.- 6.2. Further discussion of the example.- 6·3· Total number of neutrons in a family when the body is critical.- 7. Continuous time parameter.- 7.1. Integral equation treatment.- 8. Other methods.- 9. Invariance principles.- 10. One-dimensional neutron multiplication.- V. Markov branching processes (continuous time).- 1. Introduction.- 2. Markov branching processes.- 3. Equations for the probabilities.- 3.1. Existence of solutions.- 3.2. The question of uniqueness.- 3·3· A lemma.- 4. Generating functions.- 4.1. Condition that the probabilities add to 1.- 5. Iterative property of F1; the imbedded Galton-Watson process.- 5.1. Imbedded Galton-Watson processes.- 5.2. Fractional iteration.- 6. Moments.- 7. Example: the birth-and-death process.- 8. YULE’S problem.- 9. The temporally homogeneous case.- 10. Extinction probability.- 11. Asymptotic results.- 11.1. Asymptotic results when h’(1) < 1.- 11.2. Asymptotic results when h’(1) = 1.- 11.3. Asymptotic results when h’(1) > 1.- 11.4. Extensions.- 12. Stationary measures.- 13. Examples.- 13.1. The birth-and-death process.- 13.2. Another example.- 13.3. A case in which F1(1, t) < 1.- 14. Individual probabilities.- 15. Processes with several types.- 15.1. Example: the multiphase birth process.- 15.2. Chemical chain reactions.- 16. Additional topics.- 16.1. Birth-and-death processes (generalized).- 16.2. Diffusion model.- 16.3· Estimation of parameters.- 16.4. Immigration.- 16.5. Continuous state space.- 16.6. The maximum of Z (t).- Appendix 1.- Appendix 2.- VI. Age-dependent branching processes.- 1. Introduction.- 2. Family histories.- 2.1. Identification of objects in a family.- 2.2. Description of a family.- 2.3. The generations.- 3. The number of objects at a given time.- 4. The probability measure P.- 5. Sizes of the generations.- 5.1. Equivalence of {?n > 0, all n} and {Z (t) > 0, all t}; probability of extinction.- 6. Expression of Z (t, ?) as a sum of objects in subfamilies.- 7. Integral equation for the generating function.- 7.1. A special case.- 8. The point of regeneration.- 9. Construction and properties of F (s, t).- 9.1. Another sequence converging to a solution of (7.3).- 9.2. Behavior of F(0, t).- 9.3. Uniqueness.- 9.4. Another property of F.- 9.5. Calculation of the probabilities.- 10. Joint distribution of Z (t1), Z(t2),. . ., Z (tk).- 11. Markovian character of Z in the exponential case.- 12. A property of the random functions; nonincreasing character of F(1, t).- 13. Conditions for the sequel; finiteness of Z (t) and ? Z (t).- 14. Properties of the sample functions.- 15. Integral equation for M (t) = ? Z (t); monotone character of M.- 15.1. Monotone character of M..- 16. Calculation of M.- 17. Asymptotic behavior of M; the Malthusian parameter.- 18. Second moments.- 19. Mean convergence of Z (t)/n1 e?t.- 20. Functional equation for the moment-generating function of W.- 21. Probability 1 convergence of Z (t)/n1e?t.- 22. The distribution of W.- 23 · Application to colonies of bacteria.- 24. The age distribution.- 24.1. The mean age distribution.- 24.2. Stationarity of the limiting age distribution.- 24.3. The reproductive value.- 25· Convergence of the actual age distribution.- 26. Applications of the age distribution.- 26.1. The mitotic index.- 26.2. The distribution of life fractions.- 27. Age-dependent branching processes in the extended sense.- 28. Generalizations of the mathematical model.- 28.1. Transformation probabilities dependent on age.- 28.2. Correlation between sister cells.- 28.3· Multiple types.- 29. Age-dependent birth-and-death processes.- VII. Branching processes in the theory of cosmic rays (electronphoton cascades).- 1. Introduction.- 2. Assumptions concerning the electron-photon cascade.- 2.1. Approximation A.- 2.2. Approximation B.- 3. Mathematical assumptions about the functions q and k.- 3.l. Numerical values for k, q, and ?; units.- 3.2. Discussion of the cross sections.- 4. The energy of a single electron (Approximation A).- 5. Explicit representation of ? (t) in terms of jumps.- 5.1. Another expression for ? (t).- 6. Distribution of X (t) = — log ? (t) when t is small.- 7. Definition of the electron-photon cascade and of the random variable N(E, t) (Approximation A).- 7.1. Indexing of the particles.- 7.2. Histories of lives and energies.- 7.3. Probabilities in the cascade; definition of ?.- 7.4. Definition of N(E, t).- 8. Conservation of energy (Approximation A).- 9. Functional equations.- 9.1. Introduction.- 9.2. An integral equation.- 9.3. Derivation of the basic equations (11.14) in case µ = 0.- 10. Some properties of the generating functions and first moments.- 11. Derivation of functional equations for f1 and f2.- 11.1. Singling out of photons born before ?.- 11.2. Simplification of equation (11.1).- 11.3. Limiting form of f2(s, E, t + ?) as ? ? 0.- 12. Moments of N (E, t).- 12.1. First moments.- 12.2. Second and higher moments.- 12.3. Probabilities.- 12.4. Uniqueness of the solution of (11.14).- 13. The expectation process.- l3·l· The probabilities for the expectation process.- 13.2. Description of the expectation process.- 14. Distribution of Z (t) when t is large.- 14.1. Numerical calculation.- 15. Total energy in the electrons.- 15.1. Martingale property of the energy.- 16. Limiting distributions.- 16.1. Case in which t ? ?, E fixed.- 16.2. Limit theorems when t ? ? and E ? 0.- 17. The energy of an electron when ß>0 (Approximation B).- 18. The electron-photon cascade (Approximation B).- Appendix 1.- Appendix 2.