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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|Stochastic Processes

In this section, we recall some elements on stochastic processes. We fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a Banach space $H$ and a nonempty index set $\mathcal{I}$.
We begin with the following definition:
Definition 2.65. A family of (strongly $\mathcal{F}$-measurable) random variables $X \triangleq X(\cdot) \triangleq{X(t)}_{t \in \mathcal{I}}$ from $(\Omega, \mathcal{F}) \rightarrow(H, \mathcal{B}(H))$ is called an (H-valued) $s$ tochastic process. For any $\omega \in \Omega$, the map $t \mapsto X(t, \omega)$ is called a sample path (of $X$ ).

In what follows, we will choose $\mathcal{I}=[0, T]$ with $T>0$, or $\mathcal{I}=[0,+\infty)$ unless otherwise stated. Also, a stochastic process will be simply called a process if no ambiguity. An $(H$-valued) process $X(\cdot)$ is said to be continuous (resp., cádlàg, i.e., right-continuous with left limits) if there is a $\mathbb{P}$-null set $N \in \mathcal{F}$, such that for any $\omega \in \Omega \backslash N$, the sample path $X(\cdot, \omega)$ is continuous (resp., cádlàg) in $H$. In a similar way, one can define right-continuous stochastic processes, etc.

In the case that $H$ is $\mathbb{R}^{m}$ (for some $m \in \mathbb{N}$ ) with the standard topology, for a given stochastic process ${X(t)}_{t \in \mathcal{I}}$, we set
$$F_{t_{1}, \cdots, t_{j}}\left(x_{1}, \cdots, x_{j}\right) \triangleq \mathbb{P}\left(\left{X\left(t_{1}\right) \leq x_{1}, \cdots, X\left(t_{j}\right) \leq x_{j}\right}\right),$$
where $j \in \mathbb{N}, t_{i} \in \mathcal{I}, x_{i} \in \mathbb{R}^{m}$ and $X\left(t_{i}\right) \leq x_{i}$ stands for componentwise inequalities $(i=1, \cdots, j)$. Functions defined in (2.78) are called the finite dimensional distributions of $X$.

Similar to the distribution function of random variable, the finite dimensional distributions $F_{t_{1}, \cdots, t_{j}}\left(x_{1}, \cdots, x_{j}\right)$ of $X$ include the main probability properties of the process.

When $H$ is a Hilbert space, an (H-valued) stochastic process ${X(t)}_{t \in \mathcal{I}}$ is called Gaussian, if any finite linear combination $\sum_{i=1}^{k} a_{i} X\left(t_{i}\right)$ (with $a_{i} \in \mathbb{R}$ and $t_{i} \in \mathcal{I}(i=1, \cdots, k)$ and $\left.k \in \mathbb{N}\right)$ is an $H$-valued Gaussian random variable.
We shall use the following two notions in the sequel.

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

In this section, we shall introduce a special class of random variables, i.e., stopping times. We fix a filtered probability space $(\Omega, \mathcal{F}, \mathbf{F}, \mathbb{P})$ with right continuous filtration $\mathbf{F}=\left{\mathcal{F}{t}\right}{t \in[0, \infty)}$ and a Banach space $H$.

Definition 2.81. A map $\tau: \Omega \rightarrow[0, \infty]$ is called an $\mathbf{F}$-stopping time, or simply a stopping time in the case that the filtration $\mathbf{F}$ is clear from the context, if ${\tau \leq t} \in \mathcal{F}_{t}$ for any $t \geq 0$.

Obviously, when $\tau$ is a stopping time, it is a nonnegative random variable taking possibly infinite value. Define
$$\mathcal{F}{\tau}=\left{A \in \mathcal{F} \mid A \cap{\tau \leq t} \in \mathcal{F}{t}, \forall t \geq 0\right} .$$
It is clear that $\mathcal{F}{\tau}$ is a sub- $\sigma$-field of $\mathcal{F}$. The sets in $\mathcal{F}{\tau}$ can be thought of as events which may occur before the time $\tau$. The constants, i.e., $\tau(\omega) \equiv s$ $(s \geq 0)$ for every $\omega$, are stopping times and in that case $\mathcal{F}{\tau}=\mathcal{F}{s}$ (Recall that $\mathbf{F}$ can be regarded as a set of information describing the history of some phenomenon and $\mathcal{F}{s}$ is the $\sigma$-field of events prior to $s$ ). Stopping times thus appear as generalizations of constant times for each of which one can define a “past” which is consistent with the “past” of constant times. A stopping time $\tau$ is a random variable such that the event ” $\tau$ has occurred up to $t$ ” depends only on the history up to $t$, and not on any further information in the future. By means of the right-continuity of the filtration, one has Proposition 2.82. 1) A map $\tau: \Omega \rightarrow[0, \infty]$ is a stopping time if and only if ${\tau{t}$ for all $t>0$.

# 随机过程统计代考

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

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师