# 统计代写|概率论代写Probability theory代考|STAT7614

#### Doug I. Jones

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## 统计代写|概率论代写Probability theory代考|Regular Conditional Distribution

Let $X$ be a random variable with values in a measurable space $(E, \mathcal{E})$. With our machinery, so far we can define the conditional probability $\mathbf{P}[A \mid X]$ for fixed $A \in \mathcal{A}$ only. However, we would like to define for every $x \in E$ a probability measure $\mathbf{P}[\cdot \mid X=x]$ such that for any $A \in \mathcal{A}$, we have $\mathbf{P}[A \mid X]=\mathbf{P}[A \mid X=x]$ on ${X=x}$. In this section, we show how to do this.

For example, we are interested in a two-stage random experiment. At the first stage, we manipulate a coin at random such that the probability of a success (i.e., “head”) is $X$. At the second stage, we toss the coin $n$ times independently with outcomes $Y_1, \ldots, Y_n$. Hence the “conditional distribution of $\left(Y_1, \ldots, Y_n\right)$ given ${X=x}$ ” should be $\left(\operatorname{Ber}_x\right)^{\otimes n}$.

Let $X$ be as above and let $Z$ be a $\sigma(X)$-measurable real random variable. By the factorization lemma (Corollary $1.97$ with $f=X$ and $g=Z$ ), there is a map $\varphi: E \rightarrow \mathbb{R}$ such that
$\varphi$ is $\mathcal{E}-\mathcal{B}(\mathbb{R})$-measurable and $\varphi(X)=Z$
If $X$ is surjective, then $\varphi$ is determined uniquely. In this case, we denote $Z \circ X^{-1}:=$ $\varphi$ (even if the inverse map $X^{-1}$ itself does not exist).

Definition 8.24 Let $Y \in \mathcal{L}^1(\mathbf{P})$ and $X:(\Omega, \mathcal{A}) \rightarrow(E, \mathcal{E})$. We define the conditional expectation of $Y$ given $X=x$ by $\mathbf{E}[Y \mid X=x]:=\varphi(x)$, where $\varphi$ is the function from (8.10) with $Z=\mathbf{E}[Y \mid X]$.
Analogously, define $\mathbf{P}[A \mid X=x]=\mathbf{E}\left[\mathbb{1}_A \mid X=x\right]$ for $A \in \mathcal{A}$.
For a fixed set $B \in \mathcal{A}$ with $\mathbf{P}[B]>0$, the conditional probability $\mathbf{P}[\cdot \mid B]$ is a probability measure. Is this true also for $\mathbf{P}[\cdot \mid X=x]$ ? The question is a bit tricky since for every given $A \in \mathcal{A}$, the expression $\mathbf{P}[A \mid X=x]$ is defined for almost all $x$ only; that is, up to $x$ in a null set that may, however, depend on $A$. Since there are uncountably many $A \in \mathcal{A}$ in general, we could not simply unite all the exceptional sets for any $A$. However, if the $\sigma$-algebra $\mathcal{A}$ can be approximated by countably many A sufficiently well, then there is hope.

Our first task is to give precise definitions. Then we present the theorem that justifies our hope.

## 统计代写|概率论代写Probability theory代考|Processes, Filtrations, Stopping Times

We introduce the fundamental technical terms for the investigation of stochastic processes (including martingales). In order to be able to recycle the terms later in a more general context, we go for greater generality than is necessary for the treatment of martingales only.

In the following, let $(E, \tau)$ be a Polish space with Borel $\sigma$-algebra $\mathcal{E}$. Further, let $(\Omega, \mathcal{F}, \mathbf{P})$ be a probability space and let $I \subset \mathbb{R}$ be arbitrary. We are mostly interested in the cases $I=\mathbb{N}_0, I=\mathbb{Z}, I=[0, \infty)$ and $I$ an interval.

Definition $9.1$ (Stochastic process) Let $I \subset \mathbb{R}$. A family of random variables $X=$ $\left(X_t, t \in I\right)$ (on $(\Omega, \mathcal{F}, \mathbf{P})$ ) with values in $(E, \mathcal{E})$ is called a stochastic process with index set (or time set) $I$ and range $E$.

Remark $9.2$ Sometimes families of random variables with more general index sets are called stochastic processes. We come back to this with the Poisson point process in Chap. 24. $\diamond$

Remark $9.3$ Following a certain tradition, we will often denote a stochastic process by $X=\left(X_t\right)_{t \in I}$ if we want to emphasize the “time evolution” aspect rather than the formal notion of a family of random variables. Formally, both objects are of course the same. $\diamond$

Example 9.4 Let $I=\mathbb{N}0$ and let $\left(Y_n, n \in \mathbb{N}\right)$ be a family of i.i.d. Rad $1 / 2$-random variables on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$; that is, random variables with $$\mathbf{P}\left[Y_n=1\right]=\mathbf{P}\left[Y_n=-1\right]=\frac{1}{2} .$$ Let $E=\mathbb{Z}$ (with the discrete topology) and let $$X_t=\sum{n=1}^t Y_n \quad \text { for all } t \in \mathbb{N}0$$ $\left(X_t, t \in \mathbb{N}_0\right)$ is called a symmetric simple random walk on $\mathbb{Z} . \diamond$ Example $9.5$ The Poisson process $X=\left(X_t\right){t \geq 0}$ with intensity $\alpha>0$ (see Sect. 5.5) is a stochastic process with range $\mathbb{N}_0 \cdot \diamond$
We introduce some further terms.
Definition 9.6 If $X$ is a random variable (or a stochastic process), we write $\mathcal{L}[X]=$ $\mathbf{P}_X$ for the distribution of $X$. If $\mathcal{G} \subset \mathcal{F}$ is a $\sigma$-algebra, then we write $\mathcal{L}[X \mid \mathcal{G}]$ for the regular conditional distribution of $X$ given $\mathcal{G}$.

# 概率论代考

## 统计代写|概率论代写概率论代考|规则条件分布

$\varphi$是$\mathcal{E}-\mathcal{B}(\mathbb{R})$可测量的，而$\varphi(X)=Z$

$I=\mathbb{N}0$ 让 $\left(Y_n, n \in \mathbb{N}\right)$ 成为一个有身份的家庭 $1 / 2$-概率空间上的随机变量 $(\Omega, \mathcal{F}, \mathbf{P})$;也就是随机变量 $$\mathbf{P}\left[Y_n=1\right]=\mathbf{P}\left[Y_n=-1\right]=\frac{1}{2} .$$ 让 $E=\mathbb{Z}$ (离散拓扑)，让 $$X_t=\sum{n=1}^t Y_n \quad \text { for all } t \in \mathbb{N}0$$ $\left(X_t, t \in \mathbb{N}_0\right)$ 叫对称简单随机漫步吗 $\mathbb{Z} . \diamond$ 使用实例 $9.5$ 泊松过程 $X=\left(X_t\right){t \geq 0}$ 强烈地 $\alpha>0$ (见第5.5节)是一个有范围的随机过程 $\mathbb{N}_0 \cdot \diamond$我们引入一些进一步的术语。9.6 If .

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