Random gap processes and asymptotically complete sequences

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 $m\ge 2$.