Could anyone explain to me what does it mean by a map $f\to K\_f$ and $f\to \rho(f(x\_0), x\_0)$ has an algebraic tail relative some measure, where $\rho$ is Prohorov metric, from this paper [Paper](https://statweb.stanford.edu/~cgates/PERSI/papers/iterate.pdf) which are expressed in equation $(5.1)$ and $(5.2)$? 


$f\mapsto K\_f$ has algebraic tail related to $\mu\dots (5.1)$


$f\mapsto \rho(x\_0, f(x\_0)$ has algebraic tail related to $\mu\dots (5.2)$ 


They have the definition, but I am unable to relate mathematically what $(5.1)$ and $(5.2)$ will mean from the definition $(5.2)$.


A random variable $Y$ has algebraic tail if there are positive, finite constants $\alpha, \beta$ such that 
\begin{align}
 \mathbb P(Y>y)< \frac{\alpha}{y^{\beta}} 
\end{align}


Thanks a lot for any help. The idea as well as the definition I am not able to grasp, especially algebraic tail related to $\mu$.



algebraic tail of a random variable
It just means that
$$\mu\left(\{f: K\_f>y\}\right)<\frac{\alpha}{y^\beta}$$
and
$$\mu\left(\{f: \rho(x\_0,f(x\_0))>y\}\right)<\frac{\alpha}{y^\beta}$$


