Hirsch index (one of the most popular citation indices) can be in fact calculated for any finite sequence/sample of natural numbers. Indeed, let $x_0, x_1,\ldots,x_{n-1}$ be a finite sequence of natural numbers. Without loss of generality we may assume that the sequence $(x_k)$ is decreasing. Then we define the Hirsch index of $(x_k)$ as the maximal natural number $h$ such that $ f(k) = x_k \geqslant k. $ (Thus $h$ is sort of the "fixed point" of the function $f$ defined in the last display).
I wonder are there nontrivial statements/results in probability and statistics involving the Hirsch index in the sense above?