# A Continuous-Time Markov Process Taking All Possible Values

Let $$\mathbb{N}$$ be the set of positive integers. For each $$n \in \mathbb{N}$$, let $$X^{(n)}=\{ X^{(n)}(t): t \geq 0 \}$$ be a Markov chain with state-space the two point set $$\{0,1\}$$ and $$Q$$-matrix $$$$Q^{(n)} = \begin{pmatrix} - a_{n} & a_{n}\\ b_{n} & - b_{n}\\ \end{pmatrix}$$$$ where $$a_{n}, b_{n} > 0$$. Assume that $$\sum a_{n} = \infty$$ and $$\sum a_{n}/b_{n} < \infty$$.

The transition matrix is $$P^{(n)}(t) = \exp(t Q^{(n)})$$. The processes $$(X^{(n)}: n \in \mathbb{N})$$ are independent and $$X^{(n)}(0)=0$$ for every $$n$$. Each $$X^{(n)}$$ has right-continuous paths.

Consider the process $$X = (X^{(n)})$$ with values in $$\{0,1\}^{\mathbb{N}}$$. This process was introduced by David Blackwell in Another Countable Markov Process with Only Instantaneous States, Ann. Math. Stat., Volume 29 (1958), 313 - 316. His properties are studied e.g. in Kai Lai Chung, Markov Processes with Stationary Transition Probabilities or David Freedman, Markov Chains, or in the guided exercise E4.8 of David Williams, Probability with Martingales.

In a note to the last one, the author states that...

... much deeper techniques [can be used to] show that for certain choices of the sequences $$(a_{n})$$ and $$(b_{n})$$, $$X$$ will almost certainly visit every point in $$\{0,1\}^{\mathbb{N}}$$ uncountably often within a finite time.

Could someone tell me what kind of techinques he refers to? What are the conditions on $$(a_{n})$$ and $$(b_{n})$$? Is there a general theory which answers this kind of questions?

Thank you very much for your help.