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Irreducible and Aperiodic Markov Chains

Theorem 2.7   $ \;$ Let $ i\in E$ be such that $ P(X_0=i)>0$. In this case $ j$ is accessible from $ i\in E$ if and only if $ P(\tau_j <\infty\mid
X_0=i)>0$.

Proof
 

Remarks
 


Examples
 


Besides irreducibility we need a second property of the transition probabilities, namely the so-called aperiodicity, in order to characterize the ergodicity of a Markov chain in a simple way.

Definition
 


We will now show that the periods $ d_i$ and $ d_j$ coincide if the states $ i,j$ belong to the same equivalence class of communicating states. For this purpose we introduce the notation $ i\to j[n]$ if $ p_{ij}^{(n)}>0$.

Theorem 2.8   If the states $ i,j\in E$ communicate, then $ d_i=d_j$.

Proof
 


Corollary 2.5   $ \:$ Let the Markov chain $ \{X_n\}$ be irreducible. Then all states of $ \{X_n\}$ have the same period.


In order to show

Lemma 2.3   Let $ k=1,2,\ldots$ an arbitrary but fixed natural number. Then there is a natural number $ n_0\ge 1$ such that

$\displaystyle \{n_0,n_0+1,n_0+2,\ldots\}\subset\{n_1k+n_2(k+1); \
n_1,n_2\ge 0\}\,.
$

Proof
 

Theorem 2.9   $ \;$ The transition matrix $ {\mathbf{P}}$ is quasi-positive if and only if $ {\mathbf{P}}$ is irreducible and aperiodic.

Proof
 


Remarks
 


Example
$ \;$ (Diffusion Model)
see P. Brémaud (1999) Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York, p.76


Remarks
 


next up previous contents
Next: Stationary Initial Distributions Up: Ergodicity and Stationarity Previous: Estimates for the Rate   Contents
Ursa Pantle 2006-07-20