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Contrary to the common opinion, it is not true that any random walk with infinite first moment on $\mathbb Z$ is transient. Example E2 on p.87 of the second edition of Spitzer's book Principles of Ran …

No. A counterexample can be obtained in the following way. Let us begin with an integer line $\mathbb Z$. Then take a sequence of finite graphs $\Gamma_n$, and for each $n\in\mathbb Z$ attach the grap …

Actually, your idea of looking at the generation sums can be made more explicit. Let $s_n$ be the sum of $X_t$ over all vertices $t$ at a level $n\ge 0$, and let $S_n$ be the corresponding sum of $Y_t …

According to the general theory any irreducible finite state space Markov chain has a unique stationary measure, and the empirical frequencies along a.e. sample path converge to this measure. The chai …

EDIT Sorry - first mistook "transitive" for "transient". Here is an example of a simple random walk on a group. The idea, indeed, is that there is a small subset of the group near the identity with a …

For nearest neighbour random walks on certain free products the rate of escape (or, if you wish, drift) was explicitly calculated by Mairesse and Matheus. In particular, their formula (26) gives an ex …

You haven't defined what "the lazy random walk" is. Since you refer to the vertex degrees, I presume that you mean that the transition probabilities are $$ p(x,y) = \begin{cases} \frac12 \;, & \text{i …

I am not aware of any other argument. Conceptually it is actually quite simple. The main idea consists in changing the viewpoint and considering, instead of the simple random walk on a single tree, th …

The answers to all these questions are "yes" and are more or less obvious from the definitions of the corresponding properties. This is the reason why they don't appear in the literature in an explici …

This is a particular case of random walks with internal degrees of freedom (aka semi-Markov random walks or covering Markov chains). In the case when the translation group is just $\mathbb Z$ (like in …

It is always transient in dimensions 3 and higher - see Theorem T1 in Section 8 of Spitzer's classical book "Principles of random walk" (2nd ed.).

This is nothing but an ordinary Markov chain whose state space is the set of oriented edges of the graph with the transition probabilities determined by the configuration of two adjacent oriented edge …

This problem was first considered and solved by Sunada, see his 1983 paper Mean-value theorems and ergodicity of certain geodesic random walks. Alas, the authors of the quoted arxiv paper were not awa …

In my opinion, the closest to a "master theorem" is the criterion due to Terry Lyons, according to which a reversible Markov chain on a countable state space (in particular, the simple random walk on …

Unfortunately, the probabilistic argot can be sometimes not so easy to understand for outsiders. Also, I think that your notation is unnecessarily complicated. So, what do we have here? The state spa …