Ergodicity/Examples/Boolean Markov Chain

Example of Ergodicity

Let $M := \sequence {X_n}_{n \mathop \ge 0}$ be a Markov chain whose state space $S$ is a Boolean domain:

$S = \set {0, 1}$

Let $M$ be such that all the entries of the transition matrix are non-zero.

Then $M$ achieves ergodicity at the point where:

$M$ has a probability $\dfrac {p_{10} } {p_{01} + p_{10} }$ of being in state $0$

$M$ has a probability $\dfrac {p_{01} } {p_{01} + p_{10} }$ of being in state $1$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Markov chain
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Markov chain