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The Matrix of the $ n$-Step Transition Probabilities

Remarks
 

Lemma 2.1   The equation

$\displaystyle {\mathbf{P}}^{(n)}={\mathbf{P}}^n$ (22)

holds for arbitrary $ n=0,1,\ldots$ and thus for arbitrary $ n,m=0,1,\ldots$

$\displaystyle {\mathbf{P}}^{(n+m)}={\mathbf{P}}^{(n)}{\mathbf{P}}^{(m)}.$ (23)

Proof
$ \;$ Equation (22) is an immediate consequence of (20) and the definition of matrix multiplication.

$ \Box$


Example
$ \;$ (Weather Forecast)


Remarks
 

Corollary 2.2   For arbitrary $ n,m,r=0,1,\ldots$ and $ i,j,k\in E$,

$\displaystyle p^{(n+m)}_{ii}\ge p^{(n)}_{ij}p^{(m)}_{ji}$ (24)

and

$\displaystyle p^{(r+n+m)}_{ij}\ge p^{(r)}_{ik}p^{(n)}_{kk}p^{(m)}_{kj}.$ (25)


Furthermore, Lemma 2.1 allows the following representation of the distribution of $ X_n$. Recall that $ X_n$ denotes the state of the Markov chain at step $ n$.

Theorem 2.3    

Proof
 


Remarks
 

Remarks
 

Lemma 2.2    

Proof
 


next up previous contents
Next: Ergodicity and Stationarity Up: Specification of the Model Previous: Recursive Representation   Contents
Ursa Pantle 2006-07-20