Notebook for Applied Stochastic Processes
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This is a collection of my notes for SUSTech MA208 “Applied Stochastic Processes”, in the spring semester of 2022, based on Professor Zhang Yiying (张艺赢)’s lectures and slides.
毕业压力好大，暂时更新了第一章，可以作为概率论知识汇总作为参考。
- 有时间再更新整个notebook。
Chapter 1: Preliminaries: Reviews on Probability Theory
Probability Space
Definition (Probability Space).
The probability space $(\Omega,\mathcal{F},\mathbb{P})$ is defined as follows:- \(\Omega\): sample space, the set of all possible outcomes.
- \(\mathcal{F}\): \(\sigma\)-field (\(\sigma\)-algebra) on \(\Omega\) or collection of events.
- \(\mathbb{P}\): Probability measure on \((\Omega,\mathcal{F})\).
Definition ($\sigma$-algebra).
For a sample space \(\Omega\), \(\mathcal{F}\) is said to be a \(\sigma\)-algebra if it satisfies:
- \(\Omega \in \mathcal{F}\);
- closed under complement, that is, if \(E\in \mathcal{F}\), then \(E^c \in \mathcal{F}\);
- closed under countable unions of events, that is, \(E_i \in \mathcal{F}\), \(i=1,2,\cdots\), then \(\cup_iE_i \in \mathcal{F}\).
Remark: \(\mathcal{F}\) is collection of sets (For example, \(\Omega = \{a,b,c,d\}\), \(\mathcal{F}=\{\emptyset,\Omega,\{a,b\},\{c,d\}\}\)).
Definition (Probability).
Any event \(E \in \Omega\), \(\mathbb{P}(E)\) is called the probability of the event \(E\) if it satisffies:
- \(0 \leq \mathbb{P}(E) \leq 1\);
- If \(E_i \in \mathcal{F}\), \(E_i \cap E_j = \emptyset\) for \(i\neq j\) (mutually disjoint events), then \(\mathbb{P}(\cup E_i)=\sum_i\mathbb{P}(E_i)\); (概率的可数可加性)
- \(\mathbb{P}(\Omega) = 1\). (\(\mathbb{P}(\emptyset) = 0\))
Remark: \(\mathbb{P}(\cup_iE_i)\leq \sum_i\mathbb{P}(E_i)\) (subadditivitiy property or Boole’s inequality).
Remark (Continuity property of $\mathbb{P}$).
For a monotone sequence \(\{E_n,n \geq 1\}\), if and only if \(\lim\limits_{n\to \infty}\mathbb{P}(E_n)=\mathbb{P}(\lim\limits_{n \to \infty}E_n)\).
(Increasing sequence). If \(E_1 \subset E_2 \subset \cdots\), then \(\lim\limits_{n\to \infty}\mathbb{P}(E_n)=\mathbb{P}(\lim\limits_{n \to \infty}E_n)=\mathbb{P}(\cup_{n=1}^{\infty}E_n)\).
(Decreasing sequence). If \(E_1 \supset E_2 \supset \cdots\), then \(\lim\limits_{n\to \infty}\mathbb{P}(E_n)=\mathbb{P}(\lim\limits_{n \to \infty}E_n)=\mathbb{P}(\cap_{n=1}^{\infty}E_n)\).
Definition (Infimum and Supremum, 上下确界).
Suppose \(\{A_n\}_{n=1}^{\infty}\) is a sequence of sets.
Define：
\[\text{(All except finitely often, a.e.f.o.)} \quad \lim_\limits{n\to \infty}\inf A_n = \cup_{n\geq 1}\cap_{j \geq n}A_j \]
and
\[\text{(Infinitely Often, i.o.)} \quad \lim\limits_{n \to \infty}\sup A_n = \cap_{n \geq 1}\cup_{j \geq n} A_j \]
Remark:
- (a.e.f.o)
\[\begin{aligned} &x \in \lim_\limits{n\to \infty}\inf A_n = \cup_{n\geq 1}\cap_{j \geq n}A_j \newline &\Longleftrightarrow \exists n_i,x \in \cap_{j \geq n_i}A_j, \text{for some } i \newline &\Longleftrightarrow \exists n_i,x \in A_{n_i},A_{n_{i+1}}\cdots,A_{\infty} \end{aligned} \]
- (i.o.)
\[\begin{aligned}&x \in \lim_\limits{n\to \infty}\sup A_n = \cap_{n\geq 1}\cup_{j \geq n}A_j \newline &\Longleftrightarrow \forall n,x \in \cup_{j \geq n}A_j \newline &\Longleftrightarrow \forall n, \exists m\geq n, x \in A_m \end{aligned} \]
Simple Proof: (Why the infimum and superemum is defined as above?)
Let \(L_n=\cap_{j\geq n}A_j\), \(U_n = \cup_{j\geq n}A_j\), then \(L_n \leq A_n \leq U_n\).
Taking \(\lim\limits_{n\to\infty}\) we have: \(\lim\limits_{n\to\infty}L_n \leq \lim\limits_{n\to\infty}A_n \leq \lim\limits_{n\to\infty} U_n\).
Since \(\{L_n\}\) and \(\{U_n\}\) are monotonically increasing and decreasing sequences seperately, by the continuity property of \(\mathbb{P}\), we have:
\[\underbrace{\lim\limits_{n\to\infty}\cup_{n\geq1}\cap_{j\geq n}A_j}_{\lim_\limits{n\to \infty}\inf A_n} \leq \lim\limits_{n\to\infty}A_n \leq \underbrace{\lim\limits_{n\to\infty}\cap_{n\geq1}\cup_{j\geq n}A_j}_{\lim_\limits{n\to \infty}\sup A_n} \]
Proposition (Borel-Cantelli Lemma)
Let \(E_1,E_2,\cdots\) denote a sequence of events, define
\[\lim\limits_{n \to \infty}\sup E_n = \cap_{n=0}^{\infty} \cup_{i=n}^{\infty}E_i=\{\text{an infinite number of }E_i \text{ occur}\}. \]
Proposition (Borel-Cantelli Lemma I). If \(\sum_{i=1}^\infty\mathbb{P}(E_i)<\infty\), then
\[\mathbb{P}(\lim\limits_{n \to \infty}\sup E_n)=\mathbb{P}(\text{an infinite number of }E_i \text{ occur})=0. \]
Proof.
\[\begin{align} &\mathbb{P}(\lim\limits_{n \to \infty}\sup E_n) = \mathbb{P}(\cap_{n=0}^{\infty} \cup_{i=n}^{\infty}E_i) \newline &\xlongequal[]{\text{since } \cup_{i=n}^{\infty}E_i \text{ is decreasing.}} \mathbb{P}(\lim_\limits{n \to \infty}\cup_{i=n}^{\infty}E_i)=\lim_\limits{n \to \infty}\mathbb{P} (\cup_{i=n}^{\infty}E_i) \newline & \leq \lim_\limits{n \to \infty} \sum_{i=n}^{\infty}\mathbb{P}(E_i) \stackrel{(a)}{=} 0, \end{align} \]
where \((a)\) holds because of \(\sum_{i=1}^\infty\mathbb{P}(E_i)<\infty\). (Proof is straightforward by taking \(\lim\limits_{n \to \infty}\) at both sides of \(\sum_{i=1}^\infty\mathbb{P}(E_i) = \sum_{i=1}^{n-1}\mathbb{P}(E_i)+\sum_{i=n}^\infty\mathbb{P}(E_i)\).)
Proposition (Borel-Cantelli Lemma II). If \(E_1,E_2,\cdots\) are independent events such that \(\sum_{i=1}^\infty\mathbb{P}(E_i)=\infty\), then
\[\mathbb{P}(\lim\limits_{n \to \infty}\sup E_n)=\mathbb{P}(\text{an infinite number of }E_i \text{ occur})=1. \]
Proof. For any \(n\lt m\lt \infty\), since \(1-x\lt e^{-x}\),
\[\begin{align} &\mathbb{P}(\left(\cup_{i=n}^{m}E_i\right)^c)=\mathbb{P}(\cap_{i=n}^{m}E_i^c)=\prod_{i=n}^m \mathbb{P} (E_i^c) \text{ (By independence)} \newline &= \prod_{i=n}^m (1-\mathbb{P}(E_i)) \leq \prod_{i=n}^m \exp(-\mathbb{P}(E_i))=\exp\left(-\sum_{i=n}^m \mathbb{P}(E_i)\right) \stackrel{m \rightarrow \infty}{\longrightarrow} 0, \end{align} \]
which means that \(\mathbb{P}(\cup_{i=n}^{m}E_i)\to 1\) as \(m \to 0\), \(\mathbb{P}(\cup_{i=n}^{\infty}E_i)=1\) for any \(n\). Hence,
\[\mathbb{P}(\lim\limits_{n \to \infty}\sup E_n) = \mathbb{P}(\cap_{n=0}^{\infty} \cup_{i=n}^{\infty}E_i) = \mathbb{P}(\lim_\limits{n \to \infty}\cup_{i=n}^{\infty}E_i)=\lim_\limits{n \to \infty}\mathbb{P} (\cup_{i=n}^{\infty}E_i)=1. \]
- (容斥原理，Inclusion-exclusion Principle): \(A_1,A_2,\dots,A_n\) are events, then:
\[\mathbb{P}\left(\cup_{i=1}^{n} A_{i}\right)=\sum_{i=1}^{n} \mathbb{P}\left(A_{i}\right)-\sum_{i\lt j} \mathbb{P}\left(A_{i} A_{j}\right)+\sum_{i\lt j\lt k} \mathbb{P}\left(A_{i} A_{j} A_{k}\right) -\cdots+(-1)^{n+1} \mathbb{P}\left(A_{1} \cdots A_{n}\right). \]
\[\mathbb{P}(\text{exactly } r \text{ of the events } A_{1}, \cdots, A_{n}\text{ occur}) =\sum_{i=0}^{n-r}(-1)^{i}\left(\begin{array}{c} r+i \newline r \end{array}\right) \sum_{j_{1}\lt j_{2}\lt \cdots\lt j_{r+i}} \mathbb{P}\left(A_{j_{1}} A_{j_{2}} \cdots A_{j_{r+i}}\right). \]
Random Variable and Distributions
Definition (Random variable, Distribution function).
A random variable \(X\) is a mapping from \(\Omega\) to \(\mathbb{R}\), and for any Borel Set \(A\subset \mathbb{R}\) (这里等价认为，\((-\infty,a],\forall a \in \mathbb{R}\)),
\[\mathbb{P}(X(\omega)\in A) = \mathbb{P}(X^{-1}(A))=\mathbb{P}\{\omega:X(\omega)\in A\}. \]
The distribution function \(F\) of a random variable \(X\) is defined for any real number \(x \in \mathbb{R}\),
\[F(x) = \mathbb{P}(X \leq x)=\mathbb{P}(X \in (-\infty,x]), \]
and the survival (or tail) function is \(\overline{F}(x)=1-F(x)=\mathbb{P}(X > x)\).
Properties of distribution function:
- Right-continuous; Non-decreasing (\(\lim_\limits{x\to+\infty}F(x)=1,\lim_\limits{x\to-\infty}F(x)=0\));
- If \(X\) is discrete random variable, \(F(x)=\sum_{z\leq x}\mathbb{P}(x)\). If \(X\) takes uncountably many values, summation makes NO sense.
(Continuous Random Variable). \(X\) is said to be a continuous random variable if there exists a function \(f(x)\) such that for any Borel set \(B \subset \mathbb{R}\),
\[\mathbb{P}(X \subset B)=\int_Bf(x)dx. \]
\(f(x)\) is called the probability density function (pdf). For continuous r.v. \(X\),
\[F(x) = \int_{-\infty}^xf(y)dy \]
and
\[f(x) = F^{\prime} (x). \]
(Joint distribution). For random variables \(X\), \(Y\) , the joint distribution is defined by
\[F(x,y)=\mathbb{P}(X\leq x,Y \leq y). \]
Correspondingly, the distributions (marginal distributions) of \(X,Y\) are:
\[\begin{align} F_X(x)=\mathbb{P}(X\leq x)=\lim\limits_{y \to \infty}\mathbb{P}(X\leq x, Y \leq y)=\lim\limits_{y \to \infty}F(x,y) \newline F_Y(y)=\mathbb{P}(Y\leq y)=\lim\limits_{x \to \infty}\mathbb{P}(X\leq x, Y \leq y)=\lim\limits_{x \to \infty}F(x,y) \end{align} \]
Expectations
Definition (Expectation).
For a random variable \(X\), if \(\int_{-\infty}^\infty xdF(x)\) exists, then we say it is the expectation or mean of \(X\), denoted by \(\mathbb{E}[X]\).
\[\mathbb{E}[X]=\int_{-\infty}^{\infty} x d F(x)=\left\{ \begin{array}{ll}\Sigma_{x} x \mathbb{P}(X=x) & \text { if } X \text { is discrete. } \newline \int_{-\infty}^{\infty} x f(x) d x & \text { if } X \text { is continuous. }\end{array}\right. \]
Remark (Alternative expression when $X$ is nonnegative).
Let \(N\) be a nonnegative integer valued random variable, we have:
\[\mathbb{E}[N]=\sum_{k=1}^{\infty}\mathbb{P} \\{ N\geq k \\} \]
In general, if \(X\) is nonnegative with distribution \(F\), then
\[\mathbb{E}[X]=\int_0^{\infty}\overline{F}(x)dx \]
and
\[E[X^n]=\int_0^\infty nx^{n-1}\overline{F}(x)dx. \]
Proof.
\[\begin{align} E[N] &= \sum_{k=0}^{\infty}kP\{N=k\} \newline &= \sum_{k=0}^{\infty}k[P\{N \geq k\} – P\{N \geq k+1\} ] \newline &= P\{N \geq 1\} – P\{N \geq 2\} + 2\cdot P\{N \geq 2\} – 2\cdot P\{N \geq 3\} + \dots \newline &= \sum_{k=1}^{\infty} P\{N\geq k\} = \sum_{k=0}^{\infty} P\{N > k\}. \newline E[X^n] &= \int_{0}^{\infty}x^ndF(x) \newline &= \int_{0}^{\infty}\int_{0}^{x}nt^{n-1}dtdF(x) \newline &= \int_{0}^{\infty}\int_{t}^{\infty} nt^{n-1} dF(x)dt \newline &= \int_{0}^{\infty}nt^{n-1}\cdot [F(\infty) – F(t)]dt \newline &= \int_{0}^{\infty}nt^{n-1}\overline{F}(t)dt \end{align} \]
关于本博客撰写的一些统一规范
关于Mathjax结合markdown使用Latex的bug记录:
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Admonition文本框，撰写规范:
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Author Ren Zhenyu (Based on Professor Zhang Yiying's lectures)
LastMod 2024-09-22 (8da2346)