Inversive congruential pseudorandom numbers: distribution of triples

This paper deals with the inversive congruential method with power of two modulus m m for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers. It is shown that, on the average over the parameters in the inversive congruential method, the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude between m − 1 / 2 m^{-1/2} and m − 1 / 2 ( log ⁡ m ) 3 m^{-1/2}(\log m)^3 . The method of proof relies on a detailed discussion of the properties of certain exponential sums.