## 统计代写|随机过程代写stochastic process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 $\phi(\theta)=\int_{-\infty}^{\infty} e^{\theta x} d F(x)<\infty$ if $\phi(\sigma)<\infty$, where $\sigma=\operatorname{Re}(\theta)$. We also assume that $$\phi(\sigma)<\infty \text { for all } \sigma,-\beta<\sigma<\alpha<\infty, \alpha, \beta>0 .$$
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.

## 统计代写|随机过程代写stochastic process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 Markoy 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 $E\left(X_i\right)=0$.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Wald’s fundamental identit

Wald’s Sequential Analysis 提出了所谓的 Wald 标识
$$E\left(e^{\theta S_N} /[\phi(\theta)]^N\right)=1 \text { for } \phi(\theta)<\infty \text { and }|\phi(\theta)| \geq 1 .$$

$$E\left(e^{\theta S_N} z^N\right)=1+[z \phi(\theta)-1] F(z, \theta) \text { for all } \theta$$

## 统计代写|随机过程代写stochastic process代考|Fluctuation Theory

$P\left[\left|S_n-x\right|<\delta\right]>0$ 对所有人 $\delta>0$. 一个状态 $x$ 称为循环如果 $P\left[\left|S_n-X\right|<\delta\right.$ io $]=1 \mathrm{E}$ $S_n \varepsilon(x-\delta, x+\delta)$ io 概率为 1 。

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