Pruned discrete random samples

Abstract

Let X_i,i ∈ ℕ, be independent and identically distributed random variables with values in ℕ₀. We transform (`prune') the sequence {X₁,...,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.