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Generally, a cumulative process would be one in which new growth is added to old growth, and the larger the material, the more new growth.
The growth of a tree, or an avalanche, or of a snowflake, could be considered.
However, since one is generally looking at the process itself, rather than the objects on which it operates, even erosion could be considered to be a cumulative process.
In the long run, most of these processes run out of material (or the mechanism fails) and the growth of the process stops.
Perhaps this is related to what the Economist calls "the law of diminishing returns'.
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