### 数学代写|随机过程统计代写Stochastic process statistics代考|STAT3921

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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|Ergodicity

The behavior in which sample averages formed from a process converge to some underlying parameter of the process is termed ergodic. To make inference about the underlying laws governing an ergodic process, one need not observe separate independent replications of entire processes or sample paths. Instead, one need only observe a single realization of the process, but over a sufficiently long span of time. Thus, it is an important practical problem to determine conditions that lead to a stationary process being ergodic. The theory of stationary processes has a prime goal the clarification of ergodic behavior and the prediction problem for processes falling in the wide range of extremeties.

In covariance stationary process usually the added condition that $E\left(X_{t}\right)$ does not depend on $t$ is imposed. But it should be noted that in order for a stochastic process with $E\left(X_{1}^{2}\right)<\infty$ to be covariance stationary it is not necessary that its mean function $m(t)=E\left(X_{t}\right)$ be a constant. Consider the example: $X(t)=$ $\cos \left(\frac{2 \pi t}{L}\right)+Y(t)$, where $Y(t)=N(t+L)-N(t),{N(t), t \geq 0}$ be a Poisson process with intensity parameter $\lambda$ (to be defined in Chapter 7 ) and $L$ is a positive constant. Its mean function $m(t)=E\left(X_{t}\right)=\lambda(t+L)-\lambda(t)+\cos \left(\frac{2 \pi t}{L}\right)$ is functionally dependent on $t$. But \begin{aligned} \operatorname{Cov}(X(t), X(s)) &=\operatorname{Cov}(Y(t), Y(s)) \ &=\left{\begin{array}{rr} \lambda(L-|t-s|) & \text { if }|t-s| \leq L \ 0 & \text { if }|t-s|>L \end{array}\right. \end{aligned}
depends on $t-s$ only.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Exercises and Complements

Exercise 1.1 Let $a, b, c$ be independent r.v.’s uniformly distributed on $[0,1]$. What is the probability that $a x^{2}+b x+c$ has real roots?

Exercise $1.2$ Let $X$ be a Poisson r.v. with parameter $\lambda>0$. Suppose $\lambda$ itself is a r.v. following a gamma distribution with density $f(\lambda)=\frac{1}{\sqrt{n}} \lambda^{n-1} e^{-\lambda} \cdot \lambda \geq 0$. Show that $P(X=k)=\frac{\sqrt{k+n}}{\sqrt{n} \sqrt{k+1}}(1 / 2)^{k+n}, k \geq 0$ (note that when $n$ is a positive integer $X$ is negative binomial with $p=1 / 2)$.

Exercise 1.3 The following experiment is performed. An observation is made of a Poisson r.v. $X$ with parameter $\lambda$. Then a binomial event $Y$ with probability $p$ of success is repeated $X$ number of times and $Y$ successes are observed. What is the distribution of $Y$ ?

Exercise 1.4 Let $\left{X_{t}, t \geq 0\right}$ be a continuous time stochastic process with independent increments. Also $P\left(X_{0}=0\right)=1$. If $\phi(\theta, t-u)$ is the characteristic function of a single increment i.e.
$$\phi(\theta, t-u)=E\left[\exp \left(i \theta\left(X_{t}-X_{u}\right)\right)\right],$$
prove that the joint characteristic function of $X_{t_{1}}, X_{t_{2}}, \ldots, X_{t_{n}}$ where $t_{1}<t_{2}<\ldots<t_{n}$ is
$$\phi\left(\sum_{j=1}^{n} \theta_{j}, t_{1}\right) \phi\left(\sum_{j=2}^{n} \theta_{j}, t_{2}-t_{1}\right) \ldots \phi\left(\theta_{n}, t_{n}-t_{n-1}\right) .$$

Exercise 1.5 Prove that every continuous parameter Stochastic process with independent increments is a Markov process.

Exercise 1.6 Let $T$ be a nonnegative discrete random variable. Prove that $T$ has a geometric distribution iff
$$P[T>x+y \mid T>x]=P[T>y] \text { for all integers } x, y \geq 0 \text {. }$$
Exercise $1.7$ Let $T$ be a non-negative continuous random variable. Prove that $T$ has an exponential distribution iff $P[7>x+y \mid T>x]=P[T>y]$.

Exercise $1.8$ Let $T$ he a nonnegative random variable such that $X(T)$ is a stochastic process and for a fixed valuc of $T$, say $t, X(t)$ has a gamma density
$$f_{x(t)}(u)=\frac{1}{\sqrt{\alpha t}}(\lambda u)^{\alpha t-1} e^{-\lambda u}, \alpha>0, \lambda>0 .$$
Assume that the distribution of $T$ is $F(t)=P[T \leq t]$.
(a) Derive an expression for $E\left[e^{-r X(T)}\right]$
(b) Prove that $T$ has a negative binomial distribution, i.e.
$$P\left[T=\frac{\beta+k}{\alpha}\right]=\left(\begin{array}{c} \beta+k-1 \ k \end{array}\right) p^{\beta}(1-p)^{k}, \beta>0$$
$$1>p=1-q>0 . k=0.1 .2, \ldots$$
then $X(T)$ also has a gamma density. Derive its parameters.
(c) Prove that if $X(T)$ has a gamma distribution, i.e.
$E\left|e^{-w(T)}\right|=\left(\frac{\mu}{\mu+s}\right)^{\beta}$ where $\beta>0, \lambda>\mu \geq 0$, then conversely, $T$ has a negative binomial distribution. Determine its parameters.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Definition and Transition Probabilities

Here $S=$ a countable set, $T={0,1,2, \ldots},\left{X_{n}, n \geq 0\right}$ is a stochastic process satisfying $P\left[X_{n+1}=j \mid X_{0}=i_{0}, X_{1}=i_{1}, \ldots, X_{n}=i_{n}\right]=P\left[X_{n+1}=j \mid X_{n}=i_{n}\right]$, the Markov property. Then the stochastic process $\left{X_{n}, n \geq 0\right}$ is called a Markov chain (M.C.). We shall assume that the M.C. is stationary i.e. $P\left[X_{n+1}=j \mid X_{n}=\right.$ $i]=p_{i j}$ is independent of $n$ for all $i, j \in, S$. Let $P=\left(P_{i j}\right) ; i, j \in S$ be a finite or countably infinite dimensional matrix with elements $p_{i j}$.

The matrix $P$ is called the one step transition matrix of the M.C. or simply the Transition matrix or the Probability matrix of the M.C.

Example (Random Walk) A random walk on the (real) line is a Markov chain such that
$$p_{j k}=0 \text { if } k \neq j-1 \text { or } j+1 .$$
Transition is possible only to neighbouring states (from $j$ to $j-1$ and $j+1$ ). Here state space is
$$S={\ldots,-3,-2,-1,0,1,2,3, \ldots} .$$
Theorem 2.1 The Markov chain $\left{X_{n}, n \geq 0\right}$ is completely determined by the transition matrix $P$ and the initial distribution $\left{p_{k}\right}$, defined as $P\left[X_{0}=k\right]=p_{k} \geq 0$, $\sum_{k \in s} p_{k}=1$
Proof
\begin{aligned} P\left[X_{0}\right.&\left.=i_{0}, X_{1}=i_{i}, \ldots, X_{n}=i_{n}\right] \ &=P\left[X_{n}=i_{n} \mid X_{n-1}=i_{n-1}, X_{n-2}=i_{n-2}, \ldots, X_{1}=i_{1} \ldots X_{0}=i_{0}\right] \ P\left[X_{n-1}\right.&\left.=i_{n-1}, X_{n-2}=i_{n-2}, \ldots, X_{1}=i_{1}, X_{0}=i_{0}\right] \ &=P\left[X_{n}=i_{n} \mid X_{n-1}=i_{n-1}\right] P\left[X_{n-1}=i_{n-1}, \ldots, X_{0}=i_{0}\right] \ &=p_{i_{n-1} i_{n}} p_{i_{n-2} i_{n-1}} P\left[X_{n-2}=i_{n-2}, \ldots, X_{0}=i_{0}\right] \ &=p_{i_{n-1} i_{n}} p_{i_{n-2} i_{n-1}} \ldots p_{i_{i} i_{2}} p_{i_{0} i_{1}} p_{i_{0}} \text { (by induction). } \end{aligned}
Definition 2.1 A vector $u=\left(u_{1}, u_{2}, \ldots, u_{n}\right)$ is called a probability vector if the components are non-negative and their sum is one.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Exercises and Complements

φ(θ,吨−在)=和[经验⁡(一世θ(X吨−X在))],

φ(∑j=1nθj,吨1)φ(∑j=2nθj,吨2−吨1)…φ(θn,吨n−吨n−1).

FX(吨)(在)=1一个吨(λ在)一个吨−1和−λ在,一个>0,λ>0.

(a) 导出表达式和[和−rX(吨)]
(b) 证明吨具有负二项分布，即

1>p=1−q>0.ķ=0.1.2,…

(c) 证明如果X(吨)具有伽马分布，即

## 数学代写|随机过程统计代写Stochastic process statistics代考|Definition and Transition Probabilities

pjķ=0 如果 ķ≠j−1 或者 j+1.

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