  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 $m \geq 2$.
