Pruned discrete random samples

Let $X_i,i\in \mathbb{N}$, be independent and identically distributed random variables with values in ${\mathbb{N}}_0$. We transform (‘prune’) the sequence $\{X_1,\dots ,X_n\},n\in \mathbb{N}$, of discrete random samples into a sequence $\{0,1,2,\dots ,Y_n\},n\in \mathbb{N}$, of contiguous random sets by replacing $X_{n+1}$ with $Y_n+1$ if $X_{n+1}>Y_n$. We consider the asymptotic behaviour of $Y_n$ as $n\to \infty$. Applications include path growth in digital search trees and the number of tables in Pitmanʼs Chinese restaurant process if the latter is conditioned on its limit value.