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Just for interest, when sampling simplex noise in any given dimension, would the sum of all samples run towards zero?

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No, the sum will not converge (run towards zero). It may pass by zero, or be zero at some point, but there's nothing that would make the sum get closer to zero. The sum along any given dimension will be random. However, a sample of sums will tend toward a normal distribution.

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    \$\begingroup\$ The average of a bunch of samples should tend toward zero, though. In other words, although the sum won't tend toward zero, it also won't tend to grow as more samples are added. \$\endgroup\$ – Nathan Reed Dec 1 '13 at 21:38
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    \$\begingroup\$ Yep, the sum won't converge at infinity or negative infinity either. It'll just flop around forever. And I agree that the average would tend toward zero. \$\endgroup\$ – MichaelHouse Dec 1 '13 at 21:42

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