Chapter 3.2

Galton-Watson Branching Processes

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Problem 3.2.1

Find the generating function of the number of particles in the nn-th generation of a branching process, if the generating function of the number of offspring of one particle equals

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(a)

pz+1−pp z+1-p;

(b)

1−p1−pz\frac{1-p}{1-pz};

(c)

1−p(1−z)α,  α∈(0,1)1 - p(1-z)^\alpha , \; \alpha \in (0,1).

Problem 3.2.2

Find the extinction probabilities for branching processes with generating function of the number of offspring of one particle

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(a)

1−p(1−z)α,α∈(0,1)1-p(1-z)^{\alpha }, \alpha \in (0,1);

(b)

(1+z+z2+z3)/4\left(1+z+z^{2}+z^{3}\right) / 4.

Problem 3.2.3

Find the distribution of the extinction time NN for branching processes with generating function of the number of offspring of one particle

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(a)

pz+1−pp z+1-p;

(b)

1−p(1−z)α,α∈(0,1)1-p(1-z)^{\alpha }, \alpha \in (0,1)

Problem 3.2.4

A branching process has the following distribution ξ\xi of offspring of one particle:

P(ξ=0)=14,P(ξ=2)=12,P(ξ=6)=14 \mathbb {P}\left(\xi = 0\right) = \frac{1}{4}, \quad \mathbb {P}\left(\xi = 2\right) = \frac{1}{2}, \quad \mathbb {P}\left(\xi = 6\right) = \frac{1}{4}

Is it true that the extinction probability belongs to the interval (1/4,1/3)(1 / 4,1 / 3)?

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Problem 3.2.5

Let ξ\xi be the number of offspring of a particle in the Galton-Watson branching process (Xn,n∈Z+)\left(X_{n}, n \in \mathbb {Z}_{+}\right). Denote E[ξ]=μ,  Var⁡[ξ]=σ2\mathbb {E}\left[\xi \right] = \mu , \; \operatorname {Var}\left[\xi \right] = \sigma^{2}. Find E[Xn]\mathbb {E}\left[X_{n}\right] and Var⁡[Xn]\operatorname {Var}\left[X_{n}\right].

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Problem 3.2.6

Let (Xn,n∈Z+)\left(X_{n}, n \in \mathbb {Z}_{+}\right) be a branching process with offspring law ξ\xi. Denote by Yn=Xn+…+X0Y_{n} = X_{n} + \ldots + X_{0} the total number of particles in the process up to time nn, and by φYn(z)\varphi_{Y_{n}}(z) its generating function. Prove that

φYn(z)=zφξ(φYn−1(z)). \varphi _{Y_{n}}(z) = z \varphi _{\xi }\left(\varphi _{Y_{n-1}}(z)\right).
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Problem 3.2.7

Using φYn+1(z)=zφξ(φYn(z))\varphi_{Y_{n+1}}(z)=z \varphi_{\xi }\left(\varphi_{Y_{n}}(z)\right), prove that for all z∈[0,1)z \in [0,1) the limit

ρ(z)=lim⁡n→∞φYn(z), \rho (z)=\lim _{n \rightarrow \infty } \varphi _{Y_{n}}(z),

exists and satisfies the relation ρ(z)=zφξ(ρ(z))\rho (z)=z \varphi_{\xi }(\rho (z)). What value should ρ(1)\rho (1) be set to, by continuity of the function ρ\rho?

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Problem 3.2.8

Let the Galton-Watson branching process be built from a random variable ξ\xi having generating function φ(s)=1−121−s\varphi (s)=1-\frac{1}{2} \sqrt{1-s}. Find

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(a)

the extinction probability of the process;

(b)

the generating function of the total number of particles of the process;

(c)

the probability that there were a total of 10 particles in the process.