### 数学代写|随机过程统计代写Stochastic process statistics代考|MATH3801

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Wald’s Equation and Wald’s Identity

Theorem $3.2$ (Wald’s equation) Let $\left{X_{i}\right}$ be a sequence of i.i.d. r.v.s with $E(N)<\infty$. If $E\left|X_{1}\right|<\infty$ then $E\left(S_{N}\right)=\left(E X_{1}\right) E N$.

If moreover, $\sigma^{2}=\operatorname{var}\left(X_{1}\right)<\infty$, then $E\left(S_{N}-N \mu\right)^{2}=\sigma^{2} E(N)$, where $\mu=E\left(X_{1}\right)$.
Proof $E\left(S_{N}\right)=\sum_{n=1}^{\infty} E\left(S_{N} \mid N=n\right) P[N=n]$
$$=\sum_{n=1}^{\infty} \sum_{i=1}^{n} P[N=n] E\left(X_{i} \mid N=n\right)$$
$$=\sum_{i=1}^{\infty} \sum_{n=i}^{\infty} P[N=n] E\left(X_{i} \mid N=n\right)$$
(interchanging the order of summation)
$$\left|\sum_{i=1}^{\infty} \sum_{n=i}^{\infty} E\left(X_{i} \mid N=n\right) P(N=n)\right| \leq \sum_{i=1}^{\infty} \sum_{n=i}^{\infty} E\left(\left|X_{i}\right| \mid N=n\right) P(N=n)$$
$$=E\left|X_{t}\right| E(N)<\infty$$
(Fubini condition is satisfied)
Therefore
\begin{aligned} E\left(S_{N}\right) &=\sum_{i=1}^{\infty} P[N \geq i] E\left(X_{i} \mid N \geq i\right)\left(\text { since } N \geq i \text { depends on } X_{1}, \ldots, X_{i-1}\right. \text { only) }\ &=\sum_{i=1}^{\infty} P[N \geq i] E\left(X_{i}\right)=E\left(X_{i}\right) E(N) . \end{aligned}

Let $N_{n}=\min (N, n)$. Now let $N_{n} \rightarrow N$ monotonically, it follows from the Monotone convergence theorem that
$$E N_{n} \rightarrow E(N) \text { as } n \rightarrow \infty$$
Since $\left.\left{\left(S_{n}-n \mu\right)^{2}-n \sigma^{2}, S_{n}\right), n \geq 1\right}$ is a martingale (prove it).
We can apply optional sampling theorem to obtain (see Appendix iv)
$$E\left(S_{N_{n}}-n \mu\right)^{2}=\sigma^{2} E N_{n}$$
Now let $m \geq n$. Since martingales have orthogonal increments we have, by (3.7) and (3.8),
$$\begin{gathered} E\left(S_{N_{m}}-\mu N_{m}-\left(S_{N_{n}}-\mu N_{n}\right)\right)^{2}=E\left(S_{N_{m}}-\mu N_{m}\right)^{2}-E\left(S_{N_{n}}-\mu N_{n}\right)^{2} \ =\sigma^{2}\left(E N_{m}-E N_{n}\right) \rightarrow 0 \text { as } n, m \rightarrow \infty, \end{gathered}$$
that is $S_{N_{n}}-\mu N_{n}$ converges in $L_{2}$ as $n \rightarrow \infty$.
However, since we already know that $S_{N_{n}}-\mu N_{n} \rightarrow S_{N}-\mu N$ as $n \rightarrow \infty$, it follows that
$$E\left(S_{N_{n}}-\mu N_{n}\right)^{2} \rightarrow E\left(S_{N}-\mu N\right)^{2} \text { as } n \rightarrow \infty,$$
which together with (3.7) and (3.8), completes the proof.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Wald’s fundamental identit

Let $X_{1}, X_{2}, \ldots$ are i.i.d. r.v.s with $S_{n}=X_{1}+X_{2}+\ldots+X_{n}$ and $N$ is a stopping rule.

Let $F_{n}(x)=P\left[S_{n} \leq x\right], F_{1}(x)=F(x)=P\left[X_{1} \leq x\right]$ and m.g.f. of $X_{1}$ is given by
$$\phi(\theta)=\int_{-\infty}^{\infty} e^{\theta x} d F(x)<\infty \text { if } \phi(\sigma)<\infty \text {, where } \sigma=\operatorname{Re}(\theta)$$ We also assume that $$\phi(\sigma)<\infty \text { for all } \sigma,-\beta<\sigma<\alpha<\infty, \alpha, \beta>0 \text {. }$$
Under these conditions, $P\left[e^{X}<1-\delta\right]>0$ and $P\left[e^{X}>1+\delta\right]>0, \delta>0$. $\phi(\theta)$ has a minimum at $\theta=\theta_{0} \neq 0$, where $\theta_{0}$ is the root of the equation $\phi(\theta)=1 .$
Wald’s Sequential Analysis presented the so-called Wald’s identify
$$E\left(e^{\theta S_{N}} /[\phi(\theta)]^{N}\right)=1 \text { for } \phi(\theta)<\infty \text { and }|\phi(\theta)| \geq 1$$
Actually we shall give the proof of a more general theorem in Random walk due to Miller and Kemperman (1961).

Define $F_{n}(x)=P\left[S_{n} \leq x ; N \geq n\right], N=\min \left{n \mid S_{n} \notin(-b, a), 0<a, b<\infty\right}$ and the series $F(z, \theta)=\sum_{n=0}^{\infty} z^{n} \int_{-b}^{a} e^{\theta x} d F_{n}(x)$.
Then
$$E\left(e^{\theta S_{N}} z^{N}\right)=1+[z \phi(\theta)-1] F(z, \theta) \text { for all } \theta$$
which is known as Miller and Kemperman’s Identity.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Fluctuation Theory

In this section $X_{1}, X_{2}, \ldots, X_{n}, \ldots$ are i.i.d. r.v.s.
Theorem $3.3$ If $E\left|X_{i}\right|<\infty$, then \begin{aligned} P[N(b)&<\infty]=1 \text { if } E X_{i} \leq 0 \ &<1 \text { if } E X_{i}>0 \end{aligned}
For Proof see Chung and Fuchs (1951) and Chung and Ornstein (1962), Memoirs of American Math. Society.

Definition 3.2 If $S$ is uncountable, and $S_{n}=X_{1}+\ldots+X_{n}$ are Markov, $X_{i}$ ‘s being independent, then $x$ is called a possible value of the state space $S$ of the Markov chain if there exits an $n$ such that
$P\left[\left|S_{n}-x\right|<\delta\right]>0$ for all $\delta>0$. A state $x$ is called recurrent if $P\left[\left|S_{n}-X\right|<\delta\right.$ i.o. $]=1$ i.e. $S_{n} \varepsilon(x-\delta, x+\delta)$ i.o. with probability one.
We shall conclude this section by stating two very important and famous theorems whose proofs are beyond the scope of this book.
Theorem 3.4 (Chung and Fuchs)
Either every state is recurrent or no state is recurrent. (ref. Spitzer-Random Walk (1962)).
Theorem 3.5 (Chung and Ornstein)
If $E\left|X_{i}\right|<\infty$, then recurrent values exist iff $E\left(X_{i}\right)=0$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Wald’s Equation and Wald’s Identity

=∑n=1∞∑一世=1n磷[ñ=n]和(X一世∣ñ=n)

=∑一世=1∞∑n=一世∞磷[ñ=n]和(X一世∣ñ=n)
（交换求和顺序）

|∑一世=1∞∑n=一世∞和(X一世∣ñ=n)磷(ñ=n)|≤∑一世=1∞∑n=一世∞和(|X一世|∣ñ=n)磷(ñ=n)

=和|X吨|和(ñ)<∞
（满足 Fubini 条件）

## 数学代写|随机过程统计代写Stochastic process statistics代考|Wald’s fundamental identit

φ(θ)=∫−∞∞和θXdF(X)<∞ 如果 φ(σ)<∞， 在哪里 σ=回覆⁡(θ)我们还假设

φ(σ)<∞ 对所有人 σ,−b<σ<一个<∞,一个,b>0.

Wald’s Sequential Analysis 提出了所谓的 Wald 标识

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