The Theory of Branching Processes

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.