Initial point and initial distribution of the Markov chains

Question

I am reading about Markov chains on a general state space and the ergodicity theory. Some of the ergodic theorems are presented when we consider n-step transition probability conditional on initial point, sometimes we consider some initial distribution of X_{0}. I want to know if the knowledge of starting point is equivalent to knowledge of initial distribution when we consider the 'initial' behaviour ot the Markov chain? If not, is it better to consider initial ditributions or initial points of the Markov chains in statistical analysis (and ergodic theorems)?

Answer 1

Perhaps I misunderstand your question but it seems to me that you are asking about two different things. One is the set where we take measurable sets and another the measure that we attribute to each measurable set.

The first exists independently of the latter, the relation being that the support of any Markov measure (assuming that the Markov chain is stationary) is a topological Markov chain. I have to presume that you are not asking about this, in which case it is really doesn't make sense to ask about "the knowledge of starting point".

The second is the one that you describe, for which you need both the "initial" distribution and the transition probabilities, such as in $$ P(x_0,x_1)=P(x_0)P(x_1|x_0). $$