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# 数学代写|统计计算作业代写Statistical Computing代考|Markov chains

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## 数学代写|统计计算作业代写Statistical Computing代考|Discrete state space

If the state space $S$ is finite, for example $S={1,2, \ldots, N}$, then the transition probabilities $P\left(X_{n} \in A_{n} \mid X_{n-1} \in A_{n-1}\right)$ in (2.2) can be described by giving the probabilities
$$p_{x y}=P\left(X_{j}=y \mid X_{j-1}=x\right)$$
of the transitions between all pairs of elements $x, y \in S$. The resulting matrix $P=$ $\left(p_{x y}\right)_{x, y \in S}$ is called the transition matrix of the Markov chain $X$.

When considering transition matrices, it is often convenient to label the rows and columns of the matrix $P$ using elements of $S$ instead of using the usual indices ${1,2, \ldots, n}$. Thus, if $S$ is the alphabet $S={\mathrm{A}, \mathrm{B}, \ldots, \mathrm{Z}}$ we write $p_{\mathrm{AZ}}$ instead of $p_{1,26}$ to denote the probability of transitions from A to $Z$. We write $\mathbb{R}^{S \times S}$ for the set of all matrices where the columns and rows are indexed by elements of $S$. Similarly, for vectors consisting of probability weights for the elements of $S$, for example the initial distribution of a Markov chain, it is convenient to label the components of the vector by elements of $S$. We write $\mathbb{R}^{S}$ for the set of all such vectors.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Continuous state space

In this section we will briefly discuss Markov chains with continuous state space. Markov chains can be considered on very general state spaces, but here we restrict ourselves to the case $S=\mathbb{R}^{d}$. Most of the results from the previous section formally carry over to the case of continuous state space, with only changes in notation.

The concept of transition matrices in the case of continuous state space is replaced by transition kernels, as given in the following definition.
Definition 2.27 A transition kernel is a map $P(\cdot, \cdot)$ such that:
(a) $P(x, A) \geq 0$ for all $x \in \mathbb{R}^{d}$ and all $A \subseteq \mathbb{R}^{d}$; and
(b) $P(x, \cdot)$ is a probability distribution on $\mathbb{R}^{d}$ for all $x \in \mathbb{R}^{d}$.

This definition hides some of the technical complications associated with the study of Markov chains on a continuous state space. If full mathematical rigour is required, an additional condition, relating to ‘measurability’, must be included.
The idea in this definition is that $x$ takes the rôle of the current state of the Markov chain and, for given $x$, the map $A \mapsto P(x, A)$ is the distribution of the next value of the Markov chain. Thus, the transition kernel of a time-homogeneous Markov chain is defined by
$$P(x, A)=P\left(X_{j} \in A \mid X_{j-1}=x\right)$$
Often, the conditional distribution of $X_{j}$, given $X_{j-1}=x$, has a density. In this case, instead of giving a transition kernel, we can describe the transitions of a Markov chain by giving a transition density. In analogy to lemma $2.19$, a transition density is defined in the following.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|DISCRETE STATE SPACE

pX和=磷(Xj=和∣Xj−1=X)

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|CONTINUOUS STATE SPACE

b 磷(X,⋅)是一个概率分布Rd对所有人X∈Rd.