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. The main result of the paper is that if the gap distribution has a moment generating function with large enough radius of convergence, then the weight sequence is almost surely asymptotically m-complete for every \(m\ge 2\), i.e. every large enough multiple of the greatest common divisor (gcd) of gap values can be written as a sum of m distinct weights for any fixed \(m \ge 2\). Under the weaker assumption of finite \(\frac{1}{2}\)-moment for the gap distribution, we also show the simpler result that, 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.