### 统计代写|应用随机过程代写Stochastic process代考|STAT342

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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

## 统计代写|应用随机过程代写Stochastic process代考|Notion of Stochastic Processes

Loosely speaking, the mathematical description of a random phenomenon as it changes in time is a stochastic process. Since the last century there has been greater realisation that stochastic (or non-deterministic) models are more realistic than deterministic models in many situations. Observations taken at different time points rather than those taken at a fixed period of time began to draw the attention of scientists. The physicists and communication engineers played a leading role in the development of dynamic indeterminism. Many a phenomenon occurring in physical and life sciences are studied not only as a random phenomenon but also as one changing with time or space. Similar considerations are also made in other areas such as social sciences, economics and management sciences, and so on. The scope of applications of stochastic processes which are functions of time or space or both is ever increasing.

A stochastic process is a family of random variables $\left{X_{t}\right}$, where $t$ takes values in the index set $T$ (sometimes called a parameter set or a time set).
The values of $X$, are called the state space and will be denoted by $S$.
If $T$ is countable then the stochastic process is called a stochastic sequence (or discrete parameter stochastic process). If $S$ is countable then the stochastic process is called a discrete state (space) process.

If $S$ is a subset of the real line the stochastic process is called a real valued process.
If $T$ takes continuously uncountable number of values like $(0, \infty)$ or $(-\infty, \infty)$ the stochastic process is called a continuous time process. To emphasize its dependence on $t$ and sample point $w$, we shall denote the stochastic process by $X(t, w), t \in T, w \in \Omega$ i.e. for each $w \in \Omega, X_{t}=X(t$,
w) is a function of $t$.
This graph is known as the “typical sample function” or “realization of the stochastic process” $X(t, w)$.

## 统计代写|应用随机过程代写Stochastic process代考|Different Types of Stochastic Processes

Following are the most important types of stochastic processes we come across:

1. Independent stochastic sequence (Discrete time process)
$T=\lfloor 1,2,3, \ldots]$ and $\left{X_{t}, t \in T\right}$ are independent random variables.
2. Renewal process (Discrete time process)
Here $T=[0,1,2,3, \ldots], S=[0, \infty]$.
If $X_{n}$ are i.i.d. non-negative random variables and $S_{n}=X_{1}+\ldots+X_{n}$ then $\left{S_{n}\right}$ forms a discrete time (renewal process).
3. Independent increment process (Continuous time process)
$T=\left[t_{0}, \infty\right}$, where $t_{0}$ be any real number $(+$ or $-)$. For every
$$t_{0}<t_{1}<\ldots<t_{n}, t_{i} \in T, i=1,2, \ldots, n$$
if $X_{t_{0}}, X_{t_{1}}-X_{t_{0}}, X_{t_{2}}-X_{t_{1}}, \ldots, X_{t_{n}}-X_{t_{n-1}}$ are independent for all possible choices of $(1.1)$, then the stochastic process $\left{X_{t}, t \in T\right}$ is called independent increment stochastic process.
4. Markov process
\text { If } \begin{aligned} P\left[X_{t_{n+1}} \in A \mid X_{t_{n}}=a_{n}\right.&\left., X_{t_{n-1}}=a_{n-1}, \ldots, X_{t_{0}}=a_{0}\right] \ &=P\left[X_{t_{n+1}} \in A \mid X_{t_{n}}=a_{n}\right] \text { holds for all choices of } \ t_{0}<t_{1}<t_{2} &<\ldots<t_{n+1}, t_{i} \in T \cdot i=0,1,2, \ldots, n+1 \end{aligned}
and $A \in D$, the Borel field of the state space $S$, then $\left{X_{t}, t \in T\right}$ is called a Markov process.
5. Martingale or fair game process
If
$$E\left[X_{t_{n+1}} \mid X_{t_{n}}=a_{n}, X_{t_{n-1}}=a_{n-1}, \ldots, X_{t_{0}}=a_{0}\right]=a_{n}$$
i.e. $E\left[X_{t_{n+1}} \mid X_{t_{n}}, \ldots, X_{t_{0}}\right]=X_{t_{n}}$ a.s. for all choices of the partition (1.1), then $\left{X_{t}, t \in T\right}$ is called a Martingale process.

## 统计代写|应用随机过程代写Stochastic process代考|Notion of Stochastic Processes

w) 是一个函数 $t .$

## 统计代写|应用随机过程代写Stochastic process代考|Different Types of Stochastic Processes

1. 独立随机序列 (离散时间过程)
$T=\lfloor 1,2,3, \ldots]$ 和 lleft{X_{t}, t lin T\right } } \text { 是独立的随机变量。 }
2. 续订过程 (离散时间过程)
这里 $T=[0,1,2,3, \ldots], S=[0, \infty]$.
如果 $X_{n}$ 是 iid 非负随机变量和 $S_{n}=X_{1}+\ldots+X_{n} \mathrm{~ 然 㕿 业 掞 t ⿱}$ 程)。
3. 独立增量过程 (Continuous time process)
$\mathrm{T}=|$ left[t_{0}, \inftylright $}$ ， 在哪里 $t_{0}$ 是任何实数( (+或者一). 对于每一个
$$t_{0}<t_{1}<\ldots<t_{n}, t_{i} \in T, i=1,2, \ldots, n$$
如果 $X_{t_{0}}, X_{t_{1}}-X_{t_{0}}, X_{t_{2}}-X_{t_{1}}, \ldots, X_{t_{n}}-X_{t_{n-1}}$ 对于所有可能的选择都是独立的(1.1)，然后是随机 过程
4. 马尔科夫过程
If $P\left[X_{t_{n+1}} \in A \mid X_{t_{n}}=a_{n}, X_{t_{n-1}}=a_{n-1}, \ldots, X_{t_{0}}=a_{0}\right] \quad=P\left[X_{t_{n+1}} \in A \mid X_{t_{n}}=a_{n}\right]$ holds
和 $A \in D$ ，状态空间的 Borel 场 $S$ ，然后冒ft $\left{X_{-}{t}, t\right.$ in TYight $} \mathrm{~ ⿰}$
5. 鞅或公平博亦过程
If
$$E\left[X_{t_{n+1}} \mid X_{t_{n}}=a_{n}, X_{t_{n-1}}=a_{n-1}, \ldots, X_{t_{0}}=a_{0}\right]=a_{n}$$
IE $E\left[X_{t_{n+1}} \mid X_{t_{n}}, \ldots, X_{t_{0}}\right]=X_{t_{n}} \mathrm{~ 至 于 分 区 ~ ( 1 . 1 ) ~ 的 所 有 选 择 ， 那 么 攵 程 ⿰ { X _ { t } , ~ t i n ~ T V r i g h t } ~ ⿰ ⿱}$ 程。

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