# topologically recurrent Markov chain

Suppose, $$x_1, \dots, x_n, \dots$$ are random variable on $$\mathbb R$$, A Markov chain is said to be topologically recurrent if for any open set $$O$$ and starting point $$x$$, $$\mathbb P_x(\sum\limits_{n=1}^{\infty} 1_{O}(x_n)=+\infty)=1$$, where $$1_O(x_n)=1$$ if the random variable $$x_n\in O$$ and $$0$$ if not, and $$\mathbb P_x$$ is the probability for the initial value. Could anyone help me to understand the definition? I am not getting any intuition behind it. Does it saying that the Markov chain will visit every open set of $$\mathbb R$$ with probability $$1$$ after a long time i.e $$n\to \infty$$?

The series $$\sum_{n=1}^{+\infty}1_O(x_n)$$ is infinite if and only if $$\{n\mid x_n\in O\}$$ is infinite. Therefore, the notion of topologically recurrent means that for each starting point and each open set, the Markov chain will visit the open set an infinite amount of times.
• The concept seems appropriate for e.g. random walks with increments in $\mathbf R$ that are not lattice. – Olivier Aug 15 at 9:13