I am working in probability. And I am looking for an important example of action of an amenable countable group in another area of math, for which the (pointwise) ergodic theorem is actually quite important. For example I would like to know if in your area of expertize there is a group action for which probabilistic or statistical statement are of interest: e.g asymptotic of some empirical mean ect.. Thank you :)

I work in probability and I am looking for an important example of action of an amenable countable group in other areas of math for which the (pointwise) ergodic theorem is actually quite important. For example, I would like to know if in your area of expertise there is a group action for which probabilistic or statistical statements are of interest: e.g. asymptotics of some empirical mean ect.. Thank you :)