The Theory of Branching Processes

Title page
Preface
Chapter I. The Galton- Watson branching process
1. Historical remarks
2. Definition of the Galton-Watson process
2.1. Mathematical description of the Galton-Watson process
2.2. Generating functions
3. Basic assumptions
4. The generating function of Z_n
5. Moments of Z_n
6. The probability of extinction
6.1. Instability of Z_n
7. Examples
7.1. Fractional linear generating functions
7.2. Another example
7.3. Survival of animal families or genes
7.4. Electron multipliers
8. Asymptotic results when m>1
8.1. Convergence of the sequence {Z_n/m^n}
8.2. The distribution of W
8.3. Asymptotic form of P(Z_n = 0)
8.4. Local limit theorems when m > 1
8.5. Examples
9. Asymptotic results when m < 1
10. Asymptotic results when m = 1
10.1. Form of the iterates ln when m = 1
10.2. The probability of extinction when n is large
10.3. Distribution of Z_n when n is large
10.4. The past history of a surviving family when m = 1
11. Stationarity of Z_n
11.1. Stationary probabilities
11.2. Stationary measures
11.3. Existence of a stationary measure for the Galton-Watson process
11.4. The question of uniqueness
11.5. The form of the n_i when i is large, for the case m=1
11.6. Example: fractional linear generating function
12. An application of stationary measures
13. Further results on the Galton-Watson process and related topics
13.1. Joint generating function of the various generations
13.2. Distribution of Z₀+Z_l+...+Z_n and of Z = Z₀+Z_l +
13.3. The time to extinction
13.4. Estimation of parameters
13.5. Variable generating function
13.6. Trees, etc
13.7. Percolation proceses
Chapter 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
9.1. Results when ρ < 1
9.2. The case ρ = 1
9.3. Results when ρ > 1
10. Processes that are not positively regular
10.1. The total number of objects of various types
11. An exarnple from genetics
12. Remarks
12.1. Martingales
12.2. The expectation process
12.3. Fractional linear generating functions
Chapter 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 Z_n
10. Existence of eigenfunctions for M
10.1. Eigenfunctions and eigenvalues
11. Transience of Z_n
12. The case ρ =< 1
12.1. Limit theorems when ρ =< 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 sccond moment when ρ > 1
13.4. Second-order product densities
14. Convergence of Z_n/ρ^n when ρ > 1
15. Determination of the extinction probability when ρ > 1
16. Another kind of limit theorem
17. Processes with a continuous time parameter
Appendix 1
Appendix 2
Appendix 3
Chapter 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
Chapter 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 temporany 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
Chapter 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 {ζ_n>0, all n} and {Z(t)>0, ll t}; probability of extinction
6. Expression of Z(t,ω) 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(t_k)
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 EZ(t)
14. Properties of the sample functions
15. Integral equation for M(t) = EZ(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^[αt}
20. Functional equation for the moment-generating function of W
21. Probability 1 convergence of Z(t)/n₁ e^[αt}
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
Appendix
Chapter VII. Branching processes in the theory of cosmic rays (electron-photon 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
3.1. Numerical values for k, q, and λ; units
3.2. Discussion of the cross sections
4. The energy of a single electron (Approximation A)
5. Explicit representation of ε(t) in terms of jumps
5.1. Another expression for ε(t)
6. Distribution of X (t) = -log ε(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 Ω
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 bom before Δ
11.2. Simplification of equation (11.1)
11.3. Limiting form of f₂(s,E,t+Δ) as Δ goes to 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
13.1. 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 β > 0 (Approximation B)
18. The electron-photon cascade (Approximation B)
Appendix 1
Appendix 2
Bibliography
Index