Measures with superexponential moments on finitely generated groups

Question

Let $\Gamma$ be an infinite finitely generated group and let $\nu$ be a measure on $\Gamma$ which generates a transient random walk. I was reading this paper, and the authors prove many of their results under a condition which they call "superexponential moments". A measure $\nu$ is said to have superexponential moment if $$ \sum_{g \in \Gamma}c^{\| g\|}\nu(g) < \infty \text{ for all real numbers } c > 1. $$ I am somewhat familiar with the usage of "exponential moments" in literature, which is essentially if the above holds for some $c > 1$, and which follows from the measure having an exponential tail. The former is a common assumption in the study of random walks on groups. I would like to understand a bit more about the notion of "superexponential moments", for example, some history/motivation behind it, if (and where) it appears in practice (hyperbolic groups etc.), and some necessary/sufficient conditions for it. My question is mainly in the context of geometric group theory.

Answer 1

In hyperbolic context, measures with finite superexponential moments are mainly studied because of Gouezel's paper Martin boundary of random walks with unbounded jumps in hyperbolic groups, Annals of Probability 43:2374-2404, 2015.

The problem in this paper is mainly to understand to what extent results that hold for finitely supported random walks on hyperbolic groups still hold for random walks with infinite support.

In particular, the paper focuses on the Martin boundary. It has been known since the work of Ancona that for a finitely supported random walk on a hyperbolic group, the Martin boundary coincides with the Gromov boundary. But Ancona's proof crucially uses that the random walk has bounded jumps. Gouezel shows that this is still true for random walks with finite superexponential moments. On the contrary, he also shows that this is not true for random walks with only finite exponential moments. In fact, he shows that there exist uncountably many possible different Martin boundaries for measures with exponential tails, see the comments after Theorem 1.1.

This is somehow important, because it shows that the finite support assumption in Ancona's results cannot be dropped easily. This contradicts some vague feeling one could have that "results for finitely supported random walk should be generalized to random walk with large enough exponential moments". Typically for random walks on $\mathbb Z^d$ the Martin boundary identification or the local limit problem remain unchanged when going from a "finite support assumption" to "large enough exponential moments". So Gouezel shows that the situation is different for hyperbolic groups.

Now the construction of a random walk with finite superexponential moments is easy, you just need to take a superexponenial tail, but I do not know if there are classical or important examples in the context of hyperbolic group. But again, the reason for studying them in Gouezel's paper was not to consider random walks that do already exist in literature but to see what assumption should replace the finite support one.

Finally, I really think that in the paper you are refering to, the reason for mentioning such measure was to prove that the intuition that finite support can be replaced with finite superexponential moments reamins valid in relatively hyperbolic groups.

Answer 2

The Weibull distribution with exponent $k>1$ is a somewhat common distribution with finite superexponential moments. Since the linked paper says nothing relevant to your question, there seems to be hardly anything there except for the Weibull distribution.