Lower Bound
The discrepancy of arbitrary points satisfies
- ≥
for any nonzero lattice point, where denotes the number of nonzero coordinates of .
These two theorems show that the CIG is not perfect because the discrepancy is greater strictly than a positive value but also the CIG is not the worst generator as the discrepancy is lower than a value less than 1.
There exist also theorems which bound the average value of the discrepancy for Compound Inversive Generators and also ones which take values such that the discrepancy is bounded by some value depending on the parameters. For more details see the original paper.
Read more about this topic: Inversive Congruential Generator
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