An approach to nonlinear congruential methods of generating uniform pseudorandom numbers in the interval [0,1) is the Inversive congruential generator with prime modulus. A generalization for arbitrary composite moduli with arbitrary distinct primes will be present here.
Let .For integers with gcd (a,m) = 1 a generalized inversive congruential sequence of elements of is defined by
where denotes the number of positive integers less than m which are relatively prime to m.
Let take m = 15 = and . Hence and the sequence is not maximum.
The result below shows that these sequences are closely related to the following inversive congruential sequence with prime moduli.
For let and be integers with
Let be a sequence of elements of , given by
Let for be defined as above. Then
This theorem shows that an implementation of Generalized Inversive Congruential Generator is possible,where exact integer computations have to be performed only in but not in
First, observe that and hence if and only if , for which will be shown on induction on .
Recall that is assumed for . Now, suppose that and for some integer . Then straightforward calculations and Fermat's Theorem yield
which implies the desired result.
Generalized Inversive Congruential Pseudorandom Numbers are well equidistributed in one dimension. A reliable theoretical approach for assessing their statistical independence properties is based on the discrepancy of s-tuples of pseudorandom numbers.
Discrepancy bounds of the GIC Generator
We use the notation where ∈ of Generalized Inversive Congruential Pseudorandom Numbers for .
- Then the discrepancy satisfies
- < × × × for any Generalized Inversive Congruential operator.
- There exist Generalized Inversive Congruential Generators with
- ≥ × : × for all dimension s :≥ 2.
For a fixed number r of prime factors of m, Theorem 2 shows that for any Generalized Inversive Congruential Sequence. In this case Theorem 3 implies that there exist Generalized Inversive Congruential Generators having a discrepancy which is at least of the order of magnitude for all dimension . However,if m is composed only of small primes, then r can be of an order of magnitude and hence for every . Therefore, one obtains in the general case for every .
Since , similar arguments imply that in the general case the lower bound in Theorem 3 is at least of the order of magnitude for every . It is this range of magnitudes where one also finds the discrepancy of m independent and uniformly distributed random points which almost always has the order of magnitude according to the law of the iterated logarithm for discrepancies. In this sense, Generalized Inversive Congruential Pseudo-random Numbers model true random numbers very closely.
- ^ G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Clarendon Press, Oxford, 1979.
- ^ J. Kiefer, On large deviations of the empiric d.f. Fo vector chance variables and a law of the iterated logarithm, PacificJ. Math. 11(1961), pp. 649-660.
- Eichenauer-Herrmann, Jürgen (1994), On Generalized Inversive Congruential Pseudorandom Numbers (first ed.), American Mathematical Society, JSTOR 2153575