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

Random gap processes and asymptotically complete sequences