Consider a simple $1$-dimensional random walk $X_n$.

Let $p$ and $q$ (with $p+q=1$) be the probability of transitions from $n$ to $n+1$ and from $n$ to $n-1$ respectively, and suppose that $p>q$ so that the random walk is transient and with probability $1$ it will visit each site $n$ a finite number of times $V_n$.

It is not difficult to see that the law of $V_n$ conditioned on $\{V_n\geq1\}$ is geometric (using the strong Markov property), and it is even possible to find the parameter of this law and $P(V_n=0|X_0=m)$ explicitly.

My question concerns the joint law of the $(V_n)_{n\in\mathbb Z}$. It is quite clear that these random variables should not be independent, but is there any result in this sense?

For example on the joint law of $(V_n,V_{n+1})$? At least some information on their correlation?

I've had a look (a quick look, I have to admit) in the books by Feller and Spitzer, but couldn't find anything.

I would do some research on the topic myself, but I am no expert (and have no expert nearby), so I'm asking if there are classical (or known) results which I did not find.



The magic word here is `Ray-Knight Theorems'', in particular their extension to general symmetric Markov chains. A general modern reference is the book of Marcus and Rosen.


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