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…mehr
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 representationof 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.- 9. Asymptotic results when m < 1.- 10. Asymptotic results when m = 1.- 11. Stationarity of Zn.- 12. An application of stationary measures.- 13. Further results on the Galton-Watson process and related topics.- II. Processes with a finite number of types.- 1. Introduction.- 2. Definition of the multitype Galton-Watson process.- 3. The 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.- 10. Processes that are not positively regular.- 11. An example from genetics.- 12. Remarks.- III. The general branching process.- 1. Introduction.- 2. Point-distributions and set functions.- 3. Probabilities for point-distributions.- 4. Random integrals.- 5. Moment-generating functionals.- 6. Definition of the general branching process.- 7. Recurrence relation for the moment-generating functionals.- 8. Examples.- 9. First moments.- 10. Existence of eigenfunctions for M.- 11. Transience of Zn.- 12. The case ? ? 1.- 13. Second moments.- 14. Convergence of Zn/?n when ? > 1.- 15. Determination of the extinction probability when ? > 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.- 4. The first moment.-5. Criticality.- 6. Fluctuations; probability of extinction; total number in the critical case.- 7. Continuous time parameter.- 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.- 4. Generating functions.- 5. Iterative property of F1; the imbedded Galton-Watson process.- 6. Moments.- 7. Example: the birth-and-death process.- 8. YULE'S problem.- 9. The temporally homogeneous case.- 10. Extinction probability.- 11. Asymptotic results.- 12. Stationary measures.- 13. Examples.- 14. Individual probabilities.- 15. Processes with several types.- 16. Additional topics.- Appendix 1.- Appendix 2.- VI. Age-dependent branching processes.- 1. Introduction.- 2. Family histories.- 3. The number of objects at a given time.- 4. The probability measure P.- 5. Sizes of the generations.- 6. Expression of Z (t, ?) as a sum of objects in subfamilies.- 7. Integral equation for the generating function.- 8. The point of regeneration.- 9. Construction and properties of F (s, t).- 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.- 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.- 25· Convergence of the actual age distribution.- 26. Applications of the age distribution.- 27. Age-dependent branching processes in the extended sense.- 28. Generalizations of the mathematical model.- 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.- 3. Mathematical assumptions about the functions q and k.- 4. The energy of a single electron (Approximation A).- 5. Explicit representation of ? (t) in terms of jumps.- 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).- 8. Conservation of energy (Approximation A).- 9. Functional equations.- 10. Some properties of the generating functions and first moments.- 11. Derivation of functional equations for f1 and f2.- 12. Moments of N (E, t).- 13. The expectation process.- 14. Distribution of Z (t) when t is large.- 15. Total energy in the electrons.- 16. Limiting distributions.- 17. The energy of an electron when ß>0 (Approximation B).- 18. The electron-photon cascade (Approximation B).- Appendix 1.- Appendix 2.
1.- 9. Asymptotic results when m < 1.- 10. Asymptotic results when m = 1.- 11. Stationarity of Zn.- 12. An application of stationary measures.- 13. Further results on the Galton-Watson process and related topics.- II. Processes with a finite number of types.- 1. Introduction.- 2. Definition of the multitype Galton-Watson process.- 3. The 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.- 10. Processes that are not positively regular.- 11. An example from genetics.- 12. Remarks.- III. The general branching process.- 1. Introduction.- 2. Point-distributions and set functions.- 3. Probabilities for point-distributions.- 4. Random integrals.- 5. Moment-generating functionals.- 6. Definition of the general branching process.- 7. Recurrence relation for the moment-generating functionals.- 8. Examples.- 9. First moments.- 10. Existence of eigenfunctions for M.- 11. Transience of Zn.- 12. The case ? ? 1.- 13. Second moments.- 14. Convergence of Zn/?n when ? > 1.- 15. Determination of the extinction probability when ? > 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.- 4. The first moment.-5. Criticality.- 6. Fluctuations; probability of extinction; total number in the critical case.- 7. Continuous time parameter.- 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.- 4. Generating functions.- 5. Iterative property of F1; the imbedded Galton-Watson process.- 6. Moments.- 7. Example: the birth-and-death process.- 8. YULE'S problem.- 9. The temporally homogeneous case.- 10. Extinction probability.- 11. Asymptotic results.- 12. Stationary measures.- 13. Examples.- 14. Individual probabilities.- 15. Processes with several types.- 16. Additional topics.- Appendix 1.- Appendix 2.- VI. Age-dependent branching processes.- 1. Introduction.- 2. Family histories.- 3. The number of objects at a given time.- 4. The probability measure P.- 5. Sizes of the generations.- 6. Expression of Z (t, ?) as a sum of objects in subfamilies.- 7. Integral equation for the generating function.- 8. The point of regeneration.- 9. Construction and properties of F (s, t).- 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.- 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.- 25· Convergence of the actual age distribution.- 26. Applications of the age distribution.- 27. Age-dependent branching processes in the extended sense.- 28. Generalizations of the mathematical model.- 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.- 3. Mathematical assumptions about the functions q and k.- 4. The energy of a single electron (Approximation A).- 5. Explicit representation of ? (t) in terms of jumps.- 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).- 8. Conservation of energy (Approximation A).- 9. Functional equations.- 10. Some properties of the generating functions and first moments.- 11. Derivation of functional equations for f1 and f2.- 12. Moments of N (E, t).- 13. The expectation process.- 14. Distribution of Z (t) when t is large.- 15. Total energy in the electrons.- 16. Limiting distributions.- 17. The energy of an electron when ß>0 (Approximation B).- 18. The electron-photon cascade (Approximation B).- Appendix 1.- Appendix 2.
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