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 \begin{equation} Q^{(n)} = \begin{pmatrix} - a_{n} & a_{n}\\ b_{n} & - b_{n}\\ \end{pmatrix} \end{equation} 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.


1 Answer 1


I have chanced upon this old post of mine a few days ago, and I have seen that there are some people who are still interested in it. I have remembered that I wrote to the great David Williams in those days and that he kindly replied to my mail: what a fine mathematician and an exquisite person he is! By sifting through my mails of that time I could find his reply. These are his words about our issue:

On the specific point you raise about Blackwell's process, there is a curious situation. I never published a full paper on the result you mention, but a short version of the full paper I should have published appeared in the Proceedings of a NATO-funded conference. I do not have the conference booklet either, but I will do what I can to get a copy for you. I hope that there is one in the university library.
The key idea of the proof is to establish continuity of local time under suitable conditions on the a_n and b_n. This involves getting strong estimates on hitting times, etc, and then using general theory as found in the book by Blumenthal and Getoor and in some of the Strasbourg seminars.

He concluded his mail by humbly admitting that

I wrote it at a time, long ago, when I was steeped in that general-theory stuff. For 13 years, I have been trying to find a way of explaining moderately advanced Algebraic Number Theory --- an absolutely wonderful subject this --- in a less solemn way than is found in most books. So if the short version of that paper of mine contains too little information, I myself will have difficulty filling in the details.

Unfortunately neither David could find the paper among his notes, nor I could find it on Google. If someone could succeed in finding it (this is quite a challenge, since we do not even know its title, nor the year and location of this NATO-funded conference), I would be glad if he posted here its exact bibliographical reference or at least its title.


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