Pruned Discrete Random Samples

Let X i ,i ∈ ℕ, be independent and identically distributed random variables with values in ℕ0. 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 Pitman's Chinese restaurant process if the latter is conditioned on its limit value.