Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus

The inversive congruential method with modulus m = 2W for the generation of uniform pseudorandom numbers has recently been introduced. The discrepancy Z)' L of A:-tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length m/2 is the cru- cial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test. It is proved that for a pos- itive proportion of the inversive congruential generators with maximal period length, the discrepancy £>iJ2 is at least of the order of magnitude m~xl2 for all k > 2 . This shows that the bound D{2)/2 = 0(m_1/2(logw)2) established by the second author is essentially best possible.