Pruned Discrete Random Samples

Let Xi, i ∈ N, be independent and identically distributed random variables with values in N0. We transform (‘prune’) the sequence {X1, . . . , Xn}, n ∈ N, of discrete random samples into a sequence {0, 1, 2, . . . , Yn}, n ∈ N, of contiguous random sets by replacing Xn+1 with Yn + 1 if Xn+1 > Yn. We consider the asymptotic behaviour of Yn 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.