# algebraic tail of a random variable

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 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 $$\mu$$.

• care to copy in equations (5.1) and (5.2) to improve presentation and readability? Sep 4, 2019 at 21:46

It just means that $$\mu\left(\{f: K_f>y\}\right)<\frac{\alpha}{y^\beta}$$ and $$\mu\left(\{f: \rho(x_0,f(x_0))>y\}\right)<\frac{\alpha}{y^\beta}$$
• @riemann Well, "relative to $\mu$'' commonly means that we use $\mu$ as our probability measure Sep 6, 2019 at 1:26