# Limiting behaviour of Markov Chains

Is the following "vague" statement correct, and if so are there a good reference out there which formally work this out?

For a given initial distribution $$\vec{x}$$, any finite space, discrete time Markov chain described by the transition matrix $$P$$ will in the limit of many iterations $$n$$ either

1. converge to a fixed distribution $$\lim_{n \to \infty} P^{n} \vec{x}$$ (which may depend on the choice of initial distribution $$\vec{x}$$ itself).

2. or exhibit periodic behaviour with some period $$d>1$$. With periodic behaviour we mean that the state of the Markov chain "jumps" between $$d$$ different limiting distributions given by $$\{ \lim_{n \to \infty} P^{dn} \vec{x}, \lim_{n \to \infty} P^{dn+1} \vec{x}, ..., \lim_{n \to \infty} P^{dn+(d-1)} \vec{x}\}$$.

I have read through some material of Markov chains, but I never encountered such an explicit statement. Also, does a similar statement hold for general space, discrete time Markov chains?

• This is more complex because there can exist several ergodic sets, some being cyclic (with possibly different perdiods) some being regular. See Chap 2 of the classical book Finite Markov Chains by Kemeny and Snell. There are some variants in the vocabulary.
– Yves
Jan 11 '21 at 8:02