Pruned discrete random samples

Let X i , i ∈ N , be independent and identically distributed random variables with values in N 0 . We transform (‘prune’) the sequence { X 1 , … , X n } , n ∈ N , of discrete random samples into a sequence { 0 , 1 , 2 , … , Y n } , n ∈ 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 → ∞ . 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.