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# 数学代写|矩阵方法代写Applied Matrix Theory代考|MATH40550 Time Reversal

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## 数学代写|矩阵方法代写Applied Matrix Theory代考|Time Reversal

Time reversion plays an important role in the following chapters, and we shall here provide a brief account of the basic construction and properties. Consider a timehomogeneous Markov chain $\left{X_n\right}_{n \in \mathbb{N}}$ with discrete state space $E$ and transition matrix $\boldsymbol{P}=\left{p_{i j}\right}_{i, j \in E}$. Let $N>0$ be a fixed integer, and define the time-reversed process $\left{\tilde{X}n\right}{n=0, \ldots, N}$ by
$$\tilde{X}i=X{N-i}, i=0,1, \ldots, N .$$
If $\mathbb{P}\left(X_n=i\right)>0$ for all $n$ and $i \in E$, then

\begin{aligned} \mathbb{P}\left(\tilde{X}{n+1}=j \mid \tilde{X}_n=i\right) &=\mathbb{P}\left(X{N-n-1}=j \mid X_{N-n}=i\right) \ &=\frac{\mathbb{P}\left(X_{N-n}=i \mid X_{N-n-1}=j\right) \mathbb{P}\left(X_{N-n-1}=j\right)}{\mathbb{P}\left(X_{N-n}=i\right)} \ &=p_{j i} \frac{\mathbb{P}\left(X_{N-n-1}=j\right)}{\mathbb{P}\left(X_{N-n}=i\right)} . \end{aligned}

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Multidimensional Chains

Let $\left{X_i(n)\right}_{n \in \mathbb{N}}, i=1, \ldots, N$, be independent Markov chains with finite state spaces $E_i$ and transition matrices $\boldsymbol{P}i=\left{p{i: k, \ell}\right}_{k, \ell \in E_i}$. Then we form a new multidimensional process ${Y(n)}_{n \in \mathbb{N}}$ as
$$Y(n)=\left(X_1(n), \ldots, X_N(n)\right) .$$
The state space of this process is $E=E_1 \times E_2 \times \cdots \times E_N$, and the process is obviously a Markov chain. The latter follows by independence, and the Markov property of each independent process and the transition probabilities are given by
$$\mathbb{P}\left(Y(n+1)=\left(j_1, \ldots, j_N\right) \mid Y(n)=\left(i_1, \ldots, i_N\right)\right)=p_{1: i_1, j_1} p_{2: i_2, j_2} \cdots p_{N: i_N, j_N} .$$
In order to write the transition probabilities of the joint process in a more compact form, it is convenient to introduce an ordering of the state space $E$. In this way, we may consider the multidimensional process as a one-dimensional process on this larger ordered state space $E$.
A natural ordering of $n$-tuples is the lexicographical one, which is as follows.

## 数学代写|矩阵方法代写APPLIED MATRIX THEORY代考|TIME REVERSAL

$\mid$ tilde ${\mathrm{X}} \quad \mathrm{i}=X{\mathrm{Ni}}, \mathrm{i}=0,1, \backslash \mid$ dots, $\mathrm{N}$ 。

$$\mathbb{P}\left(\tilde{X} n+1=j \mid \tilde{X}n=i\right)=\mathbb{P}\left(X N-n-1=j \mid X{N-n}=i\right) \quad=\frac{\mathbb{P}\left(X_{N-n}=i \mid X_{N-n-1}=j\right) \mathbb{P}\left(X_{N-n-1}=j\right)}{\mathbb{P}\left(X_{N-n}=i\right)} \frac{\mathbb{P}\left(X_{N-n-1}=j\right)}{\mathbb{P}\left(X_{N-n}=i\right)} .$$

## 数学代写|矩阵方法代写APPLIED MATRIX THEORY代 考|MULTIDIMENSIONAL CHAINS

. Thenweformanewmultidimensionalprocess ${$ 是 $n}$ _ ${n \backslash$ in $\backslash$ mathbb ${\mathrm{N}]} a s Y(n)=\left(X_1(n), \ldots, X_N(n)\right)$. Thestatespaceofthisprocessis $\mathrm{E}=\mathrm{E}{-} 1 \backslash$ thimes $\mathrm{E}{-} 2$
|times \cdots \times E_N
, andtheprocessisobviouslya Markovchain. Thelatter followsbyindependence, andthe Markovpropertyofeachindependentprocessandthetransitionpr
$\mathbb{P}\left(Y(n+1)=\left(j_1, \ldots, j_N\right) \mid Y(n)=\left(i_1, \ldots, i_N\right)\right)=p_{1: i_1 j_1} p_{2: i_2, j_2} \cdots p_{N: i_N, j_N}$.
Inordertowritethetransitionprobabilitiesofthejointprocessinamorecompact form, itisconvenienttointroduceanorderingofthestatespace 和

• Inthisway, wemayconsiderthemultidimensionalprocessasaone – dimensionalprocessonthislargerorderedstatespace 和. Anaturalorderingof
$n \$\$-tuples 是字典式的，如下所示。

## Matlab代写

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