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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.

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. 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, and some necessary/sufficient conditions for it.

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.

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Measures with superexponential moments on finitely generated groups

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. 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, and some necessary/sufficient conditions for it.