Random Gap Processes and Asymptotically Complete Sequences

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