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# 数学代写| 随机过程代考|Martingales

A population starts with one individual at time $n=0: Z_{0}=1$.

After one unit of time (at time $n=1$ ) the sole individual produces $Z_{1}$ identical clones of itself and dies. $Z_{1}$ is an $\mathbb{N}_{0}$-valued random variable.

(a) If $Z_{1}$ happens to be equal to 0 the population is dead and nothing happens at any future time $n \geq 2$.

(b) If $Z_{1}>0$, a unit of time later, each of $Z_{1}$ individuals gives birth to a random number of children and dies. The first one has $Z_{1,1}$ children, the second one $Z_{1,2}$ children, etc. The last, $Z_{1}^{\text {th }}$ one, gives birth to $Z_{1, Z_{1}}$ children. We assume that the distribution of the number of children is the same for each individual in every generation and independent of either the number of individuals in the generation and of the number of children the others have. This distribution, shared by all $Z_{n, i}$ and $Z_{1}$, is called the offspring distribution. The total number of individuals in the second generation is now
$$Z_{2}=\sum_{k=1}^{Z_{1}} Z_{1, k}$$
(c) The third, fourth, etc. generations are produced in the same way. If it ever happens that $Z_{n}=0$, for some $n$, then $Z_{m}=0$ for all $m \geq n$ – the population is extinct. Otherwise,
$$Z_{n+1}=\sum_{k=1}^{Z_{n}} Z_{n, k}$$

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## 数学代写| 随机过程代考|martingale

Definition. A sequence $\left{S_{n}: n \geq 1\right}$ is a martingale with respect to the sequence $\left{X_{n}: n \geq 1\right}$ if, for all $n \geq 1:$
(a) $\mathbb{E}\left|S_{n}\right|<\infty$,
(b) $\mathbb{E}\left(S_{n+1} \mid X_{1}, X_{2}, \ldots, X_{n}\right)=S_{n}$.
Equation (2) shows that the sequence of gambler’s fortunes is a martingale with respect to itself. The extra generality, introduced by the sequence $\left{X_{n}\right}$ in (3), is useful for martingales which arise in the following way. A specified sequence $\left{X_{n}\right}$ of random variables, such as a Markov chain, may itself not be a martingale. However, it is often possible to find some function $\phi$ such that $\left{S_{n}=\phi\left(X_{n}\right): n \geq 1\right}$ is a martingale. In this case, the martingale property (2) becomes the assertion that, given the values of $X_{1}, X_{2}, \ldots, X_{n}$, the mean value of $S_{n+1}=\phi\left(X_{n+1}\right)$ is just $S_{n}=\phi\left(X_{n}\right)$; that is,

## 数学代写| 随机过程代考|Branching processes

Branching processes, two martingales. Let $Z_{n}$ be the size of the $n$th generation of a branching process, with $Z_{0}=1$. Recall that the probability $\eta$ that the process ultimately becomes extinct is the smallest non-negative root of the equation $s=G(s)$, where $G$ is the probability generating function of $Z_{1}$. There are (at least) two martingales associated with the process. First, conditional on $Z_{n}=z_{n}, Z_{n+1}$ is the sum of $z_{n}$ independent family sizes, and so

## 数学代写| 随机过程代考|MARTINGALE

b 和(小号n+1∣X1,X2,…,Xn)=小号n.

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