Math Problems Vault

Problem 35.1

Let $\xi_{0}, \xi_{1}, \ldots$ be a branching process, where $\xi_{0} = 1$ and $\mathbb {E}\left[\xi_{1}\right] = m$ . Find $\mathbb {E}\left[\xi_{n+k} \mid \xi_{n}\right]$ .

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Problem 35.2

Find the generating function of the number of particles in the $n$ -th generation, if the generating function of the offspring of a single particle equals $pz + 1 - p$ .

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Problem 35.3

(a)(b)

Find the generating function of the number of particles in the second generation, if the generating function of the offspring of a single particle equals:

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(a)

$1 - p(1-z)^{\alpha }$ , $0 < \alpha < 1$ ;

(b)

$(1-p)/(1-pz)$ .

Problem 35.4

Find the extinction probability of a branching process, if the generating function of the number of offspring of a single particle equals:

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(a)

$pz + 1 - p$ ;

(b)

$1 - p(1-z)^{\alpha }$ , $0 < \alpha < 1$ ;

(c)

$(1-p)/(1-pz)$ ;

(d)

$1 - p + pz^{2}$ ;

(e)

$(1+z+z^{2})/3$ .

Problem 35.5

In problem 35.4(a)--(c), find the distribution of the extinction time of the branching process.

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Problem 35.6

Let $\xi_{n}$ be a branching process with initial population size $N$ and generating function $\phi (z) = 1-p+pz$ . Find the distribution of the extinction time $\tau$ of the process.

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Problem 35.7

Let $\xi_{0}, \xi_{1}, \ldots$ be a branching process, $\xi_{0} = 1$ . Prove that

\mathbb {P}\left(\xi _{n} > N \text{ for some } 1 \leq n \leq m-1 \mid \xi _{m} = 0\right) \leq \mathbb {P}_{N}\left(\xi _{m} = 0\right).

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Problem 35.8

Find the generating function of the total number of particles in the first $n$ generations, if the generating function of the immediate offspring of a single particle equals $pz + 1 - p$ .

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Problem 35.9

Let a particle have $k$ direct offspring with probability $p_{k} = bc^{k-1}$ , $k=1,2,\ldots$ , and $p_{0} = 1 - p_{1} - p_{2} - \ldots$ , where $b, c > 0$ and $b + c < 1$ . Find the generating function of the immediate offspring of a single particle.

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Problem 35.10

Consider a branching process $\xi_{0}, \xi_{1}, \ldots$ with generating function

\phi (z) = \frac{1-b-c}{1-c} + \frac{bz}{1-cz},

where $0 < c < b+c < 1$ and $1-b-c > c(1-c)$ . Find

\lim _{n \to \infty } \mathbb {P}\left(\xi _{n} = k \mid \xi _{n} > 0\right).

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Problem 35.11

Let, in the branching process $\xi_{0}, \xi_{1}, \ldots$ of problem 35.10, the equality $1-b-c = c(1-c)$ hold. Find

\lim _{n \to \infty } \mathbb {P}\left(\xi _{n} \leq nx \mid \xi _{n} > 0\right).

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