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

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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|Key concepts in stochastic processes

Stochastic processes model systems that evolve randomly in time, space or spacetime. This evolution will be described through an index $t \in T$. Consider a random experiment with sample space $\Omega$, endowed with a $\sigma$-algebra $\mathcal{F}$ and a base probability measure $P$. Associating numerical values with the elements of that space, we may define a family of random variables $\left{X_{t}, t \in T\right}$, which will be a stochastic process. This idea is formalized in our first definition that covers our object of interest in this book.

Definition 1.1: A stochastic process $\left{X_{t}, t \in T\right}$ is a collection of random variables $X_{t}$, indexed by a set $T$, taking values in a common measurable space $S$ endowed with an appropriate $\sigma$-algebra.
$T$ could be a set of times, when we have a temporal stochastic process; a set of spatial coordinates, when we have a spatial process; or a set of both time and spatial coordinates, when we deal with a spatio-temporal process. In this book, in general,

we shall focus on stochastic processes indexed by time, and will call $T$ the space of times. When $T$ is discrete, we shall say that the process is in discrete time and will denote time through $n$ and represent the process through $\left{X_{n}, n=0,1,2, \ldots\right}$. When $T$ is continuous, we shall say that the process is in continuous time. We shall usually assume that $T=[0, \infty)$ in this case. The values adopted by the process will be called the states of the process and will belong to the state space $S$. Again, $S$ may be either discrete or continuous.

At least two visions of a stochastic process can be given. First, for each $\omega \in \Omega$, we may rewrite $X_{t}(\omega)=g_{\omega}(t)$ and we have a function of $t$ which is a realization or a sample function of the stochastic process and describes a possible evolution of the process through time. Second, for any given $t, X_{t}$ is a random variable. To completely describe the stochastic process, we need a joint description of the family of random variables $\left{X_{t}, t \in T\right}$, not just the individual random variables. To do this, we may provide a description based on the joint distribution of the random variables at any discrete subset of times, that is, for any $\left{t_{1}, \ldots, t_{n}\right}$ with $t_{1}<\cdots<t_{n}$, and for any $\left{x_{1}, \ldots, x_{n}\right}$, we provide
$$P\left(X_{t_{1}} \leq x_{1}, \ldots, X_{t_{n}} \leq x_{n}\right)$$
Appropriate consistency conditions over these finite-dimensional families of distributions will ensure the definition of the stochastic process, via the Kolmogorov extension theorem, as in, for example, Øksendal (2003).

## 统计代写|应用随机过程代写Stochastic process代考|Main classes of stochastic processes

Except for the case of independence, the simplest dependence form among the random variables in a stochastic process is the Markovian one.

Definition 1.6: Consider a set of time instants $\left{t_{0}, t_{1}, \ldots, t_{n}, t\right}$ with $t_{0}<t_{1}<\cdots<$ $t_{n}<t$ and $t, t_{i} \in T$. A stochastic process $\left{X_{t}, t \in T\right}$ is Markovian if the distribution

of $X_{t}$ conditional on the values of $X_{t_{1}}, \ldots, X_{t_{n}}$ depends only on $X_{t_{n}}$, that is, the most recent known value of the process
$$\begin{gathered} P\left(X_{t} \leq x \mid X_{t_{x}} \leq x_{n}, X_{t_{n-1}} \leq x_{n-1}, \ldots, X_{t_{0}} \leq x_{0}\right) \ =P\left(X_{t} \leq x \mid X_{t_{n}} \leq x_{n}\right)=F\left(x_{n}, x ; t_{n}, t\right) \end{gathered}$$
As a consequence of the previous relation, we have
$$F\left(x_{0}, x ; t_{0}, t_{0}+t\right)=\int_{y \in S} F(y, x ; \tau, t) \mathrm{d} F\left(x_{0}, y ; t_{0}, \tau\right)$$
with $t_{0}<\taun_{1}>\cdots>n_{k}$, we have
\begin{aligned} P\left(X_{n}=j \mid X_{n_{1}}=i_{1}, X_{n_{2}}=i_{2}, \ldots, X_{n_{k}}=i_{n_{k}}\right) &=\ P\left(X_{n}=j \mid X_{n_{1}}=i_{1}\right) &=p_{i_{1} j}^{\left(n_{1}, n\right)} \end{aligned}
Using this property and taking $r$ such that $m<r<n$, we have
\begin{aligned} p_{i j}^{(m, n)} &=P\left(X_{n}=j \mid X_{m}=i\right) \ &=\sum_{k \in S} P\left(X_{n}=j \mid X_{r}=k\right) P\left(X_{r}=k \mid X_{m}=i\right) \end{aligned}
Equations (1.4) and (1.5) are called the Chapman-Kolmogorov equations for the continuous and discrete cases, respectively. In this book we shall refer to discrete state space Markov processes as Markov chains and will use the term Markov process to refer to processes with continuous state spaces and the Markovian property.

## 统计代写|应用随机过程代写Stochastic process代考|Discrete time Markov chains

Markov chains with discrete time space are an important class of stochastic processes whose analysis serves as a guide to the study of other more complex processes. The main features of such chains are outlined in the following text. Their full analysis is provided in Chapter 3 .

Consider a discrete state space Markov chain, $\left{X_{n}\right}$. Let $p_{i j}^{(m, n)}$ be defined as in (1.5), being the probability that the process is at time $n$ in $j$, when it was in $i$ at time $m$. If $n=m+1$, we have
$$p_{i j}^{(m, m+1)}=P\left(X_{m+1}=j \mid X_{m}=i\right)$$
which is known as the one-step transition probability. When $p_{i j}^{(m, m+1)}$ is independent of $m$, the process is stationary and the chain is called time homogeneous. Otherwise,

the process is called time inhomogeneous. Using the notation
\begin{aligned} &p_{i j}=P\left(X_{m+1}=j \mid X_{m}=i\right) \ &p_{i j}^{n}=P\left(X_{n+m}=j \mid X_{m}=i\right) \end{aligned}
for every $m$, the Chapman-Kolmogorov equations are now
$$p_{i j}^{n+m}=\sum_{k \in S} p_{i k}^{n} p_{k j}^{m}$$
for every $n, m \geq 0$ and $i, j$. The $n$-step transition probability matrix is defined as $\mathbf{P}^{(n)}$, with elements $p_{i j}^{n}$. Equation (1.6) is written $\mathbf{P}^{(n+m)}=\mathbf{P}^{(n)} \cdot \mathbf{P}^{(m)}$. These matrices fully characterize the transition behavior of an homogeneous Markov chain. When $n=1$, we shall usually write $\mathbf{P}$ instead of $\mathbf{P}^{(1)}$ and shall refer to the transition matrix instead of the one-step transition matrix.

## 统计代写|应用随机过程代写Stochastic process代考|Main classes of stochastic processes

F(X0,X;吨0,吨0+吨)=∫是∈小号F(是,X;τ,吨)dF(X0,是;吨0,τ)

p一世j(米,n)=磷(Xn=j∣X米=一世) =∑ķ∈小号磷(Xn=j∣Xr=ķ)磷(Xr=ķ∣X米=一世)

## 统计代写|应用随机过程代写Stochastic process代考|Discrete time Markov chains

p一世j(米,米+1)=磷(X米+1=j∣X米=一世)

p一世j=磷(X米+1=j∣X米=一世) p一世jn=磷(Xn+米=j∣X米=一世)

p一世jn+米=∑ķ∈小号p一世ķnpķj米

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