Largest-first sequential selection with a sum constraint

Abstract

Scan the order statistics X (1) ⩾ X (2) ⩾ … ⩾ X ( n) of n independent samples from U[0, 1] in the order listed. The i-th number scanned, X ( i) , is selected if and only if the sum of X ( i) and the numbers already selected out of the first i −1 is no greater than 1. Let N( n) denote the total number selected. We prove that N∞  lim n→∞ E[ N( n)] = 1 + Σ k⩾1 E[ N( k)]/2 k+1 > 5/3, and that β N ∞ - E[ N( n)] β </ n2 − n . The 5 3 bound is close; numerical evaluation of a recursive formula shows that N ∞ = 1.64659337± 3 × 10 −8. Similar convergence results are obtained for the function W( n) = 1 − S( n), where S( n) is the sum of the items selected.