# What is the ergodic Markov chain

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**content**

Basic definitions and quasi-positive transition matrices

This leads to the following concept of *Ergodicity* of Markov chains.

**definition**- The Markov chain with the transition matrix or the associated -stepped transition matrices ) called
*ergodic*if the limits(33) - for each exist,
- are positive and not of depend,
- a probability function form, i.e. .

**example**- (
*Weather forecast*)

The ergodicity of Markov chains with general (finite) state space can be characterized with the help of the following term from the theory of positive matrices.

**definition****Notice**- If is a stochastic matrix for which it is a natural number there, so that all entries of are positive, then all entries are also from For
*each*natural number positive.

**proof**-
- We first show that the condition
(35)

for a sufficient for the ergodicity of is.- Be and . The Chapman-Kolmogorov equation (23) then shows that and thus i.e., for each , where we put. In the same way it turns out that for each applies.
- To the existence of the limit values To prove in (33), it suffices to show that for each
(36) - For this we consider for arbitrary, but firmly given states the amounts and .
- Be . Then applies and
- By applying the Chapman-Kolmogorov equation (23) again, it follows that for any and
- It follows that and (by induction) that for each
(37) - So there is an (infinitely growing) sequence of natural numbers such that for each
(38) - Because the differences monotonous in (38) holds for
*each*(infinitely growing) sequence of natural numbers. - This proves the validity of (36).

- The limit values are positive because
- Also applies , because the interchange of limit and
*finite*Summation is allowed. - The necessity of condition (35) follows directly from this and (33), taking into account that the state space is finite.

- We first show that the condition
**Notice**-
- Because the limits Ergodic Markov chains not from depend, results from this and from the finiteness of the state space that
- In the proof of Theorem 2.4, not only was the existence of the limit values shown, but the following
*Estimation of the speed of convergence*derived: From (37) it follows that(39)

or.(40)

in which the integer part of designated. - In connection with (39) and (40) one speaks in the literature of
*geometric bounds*the speed of convergence.

We now show that the limit values can also be understood as the solution of a linear system of equations.

**proof****Notice**

**Next page:**Estimation of the speed of convergence; Perron-Frobenius theorem

**Upwards:**Ergodicity and Stationarity

**Previous page:**Ergodicity and Stationarity & nbsp

**content**Ursa Pantle 2003-09-29

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