Fluctuations of ergodic averages for amenable group actions

We show that for any countable amenable group action, along Folner sequences have $c>1$ a two sided $c$-tempered tail, one universal estimate the probability there are $n$ fluctuations in ergodic averages of $L^{\infty}$ functions, and this gives exponential decay $n$. Any two-sided sequence can be thinned out to satisfy above property, particular, amenble admits such sequence. This extends results S. Kalikow B. Weiss $\mathbb{Z}^{d}$ actions N. Moriakov groups with polynomial growth.