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.

我们研究了生成随机正整数权重序列\（\ {W_n \} \）的过程，其中权重\（\ {X_n = W_n-W_ {n-1} \} \）之间的差距是正整数值随机变量。本文的主要结果是，如果间隙分布具有具有足够大的会聚半径的矩生成函数，则对于每个\（m \ ge 2 \），即每个足够大的加权序列，几乎可以肯定地渐近m -complete间隙值的最大公因数（gcd）的倍数可以写成任何固定\（m \ ge 2 \）的m个不同权重的总和。在有限\（\ frac {1} {2} \）的较弱假设下对于间隙分布的矩，我们还显示了一个更简单的结果，即几乎可以肯定地，所得的权重序列是渐近完整的，即，可能间隙值的gcd的所有足够大的倍数都可以写为不同权重的总和。