数学代写|ST302 Stochastic processes

Consider the unit interval $I:=[0,1]$. Moreover, for every $n$ and $\left(i_1, \ldots, i_n\right) \in{0,1,2}^n$ we consider the interval $I_{i_1 \ldots i_n} \subset I$ which is the set of those numbers whose base 3 expansion starts with $\left(i_1 \ldots i_n\right)$. That is
$$
I_{i_1 \ldots i_n}:=\left[\sum_{k=1}^n \frac{i_k}{3^k}, \frac{1}{3^n}+\sum_{k=1}^n \frac{i_k}{3^k}\right] .
$$
Let $X_0, X_1, X_2$ be independent $\operatorname{Bernoulli}\left(p_0\right), \operatorname{Bernoulli}\left(p_1\right)$ and $\operatorname{Bernoulli}\left(p_2\right)$ random variables respectively. That is $\mathbb{P}\left(X_i=1\right)=p_i$ and $\mathbb{P}\left(X_i=0\right)=1-p_i$ for $i=0,1,2$. Moreover for every $n$ and $\left(i_1, \ldots, i_n\right) \in{0,1,2}^n$ we are given the random variables $X_{i_1 \ldots i_n}$ such that on the one hand $\left{X_{i_1 \ldots i_n}\right}_{n \geq 1,\left(i_1, \ldots, i_n\right) \in{0,1,2}^n}$ are independent and on the other hand: $X_{i_1 \ldots i_n} \stackrel{d}{=} X_{i_n}$. For every $n \geq 1$ we define the set $E_n \subset[0,1]$ by
$$
E_n:=\bigcup_{X_{i_1} \cdot X_{i_1, i_2} \cdots X_{i_1, i_2, \ldots, i_n}=1} I_{i_1 \ldots i_n} .
$$
Finally, we define the set $E:=\bigcap_{n=1}^{\infty} E_n$. Assume that $p_0=\frac{2}{3}, p_1=\frac{3}{4}$ and $p_2=\frac{1}{2}$. Question: Is it true that $\mathbb{P}(E \neq \emptyset)>0$

Given a branching process with the following offspring distribution, determine the extinction probabilities $q$ :
(a) $p_0=0.25, p_1=0.4, p_2=0.35, p_n=0$ if $n \geq 3$,
(b) $p_0=0.5, p_1=0.1, p_2=0, p_3=0.4, p_n=0$ if $n \geq 4$.

Consider the branching process with offspring distribution as in the previous exercise part (b). What is the probability that the population is extinct in the second generation $X_2=0$, given that it did not die out in the first generation?

Consider the branching process with offspring distribution given by $\left{p_n\right}_{n=0}^{\infty}$. We change this process into an irreducible Markov chain by the following modification:
whenever the population dies out, then the next generation has exactly one new individual. That is $\mathbb{P}\left(X_{n+1}=1 \mid X_n=0\right)=p(0,1)=1$. For which $\left{p_n\right}_{n=0}^{\infty}$ will this chain be null recurrent, recurrent, transient?

Let $X_1, X_2, \ldots$ i.i.d. random variables taking values in the integers such that $\mathbb{E}\left[X_i\right]=0$ for all $i$. Let $S_0:=0$ and $S_n:=X_1+\cdots+X_n$.
(a) Let $G_n(x):=\sum_{j=0}^n \mathbb{1}{\left{S_j=x\right}}$. That is $G_n(x)$ is the expected number of visits to $x$ in the first $n$ steps. Show that for all $n$ and $x, G_n(0) \geq G_n(x)$. (Hint: consider the first $j$ with $S_j=x$.) (b) Note that the Law of Large Numbers implies that for each $\varepsilon>0$ we have: $\lim {n \rightarrow \infty} \mathbb{P}\left(\left|S_n\right| \leq n \varepsilon\right)=1$. Using this prove that for each $\varepsilon>0$ we have $\lim {n \rightarrow \infty} \frac{1}{n} \sum{|x| \leq \varepsilon \cdot n, x \in \mathbb{Z}} G_n(x)=1$
(c) Using (a) and (b), show that for each $M<\infty$ there is an $n$ such that $G_n(0) \geq M$.
(d) Now prove that $S_n$ is a recurrent Markov chain.