# When doesn't a Markov Chain have a stationary distribution?

Is it possible for a Markov Chain not to have a stationary distribution?

When doesn't a Markov Chain have a stationary distribution?

• The only possibility is when the state space is infinite. There is always at least one stationary distribution for any finite state Markov chain. If the chain is irreducible then the stationary distribution is unique. – Michael May 7 at 0:01

A Markov chain that wanders off to $$\infty$$ like an asymmetric random walk does not have a stationary distribution.
• Let $X_n=Y_1+Y_2+...+Y_n$ where $Y_n$'s are i.i.d taking values $+1$ and $-1$ with probabilities $3/4$ and $1/4$. Intuitively you move one step at a time with higher probability of moving to the right. It is well known that $X_n \to \infty$ and there is no stationary distribution. – Kabo Murphy May 6 at 23:36