Random gap processes and asymptotically complete sequences

TL;DR

This paper investigates random integer weight sequences generated by i.i.d. gaps, proving asymptotic completeness under finite half-moment conditions and stronger sum representation results with certain moment generating function conditions.

Contribution

It introduces new conditions under which random weight sequences are asymptotically complete and can represent large multiples of the gcd as sums of a fixed number of distinct weights.

Findings

Asymptotic completeness holds if the gap distribution has finite 1/2-moment.

Stronger sum representation results are obtained when the gap distribution's moment generating function has a large radius of convergence.

Large enough multiples of the gcd can be expressed as sums of a fixed number of distinct weights.

Abstract

We study a process of generating random positive integer weight sequences $\{ W_n \}$ where the gaps between the weights $\{ X_n = W_n - W_{n-1} \}$ are i.i.d. positive integer-valued random variables. We show that as long as the gap distribution has finite $\frac{1}{2}$-moment, almost surely, the resulting weight sequence is asymptotically complete, i.e., all large enough multiples of the gcd of the possible gap values can be written as a sum of distinct weights. We then show a much stronger result that if the gap distribution has a moment generating function with large enough radius of convergence, then every large enough multiple of the gcd of gap values can be written as a sum of $m$ distinct weights for any fixed $m \geq 2$.