Statistical independence properties of pseudorandom vectors produced by matrix generators (Q920538)

For a short general discussion of the generators see p. 87 of [\textit{P. L'Écuyer}: Random numbers for simulation, CACM 33, 86 ff (1990)] who also explains discrepancy (informally) and its importance. In the present publication the tests for one-dimensional generators are generalised to k-vectors of random numbers from multiplicative congruential generators with constant square matrix factor and a common prime p as modulus. The analysis requires recourse to some number theory in order to get results characteristic of the powers of a matrix over the finite field of order p. Upper and lower bounds for the discrepancy are derived, the latter one by introducing a condensing ``figure of merit'' analog to the one- dimensional case. With these results matrices as factors with good statistical behaviour may be constructed for any dimension depending on the ``lag''s (length of series of vectors considered for discrepancy, i.e. for mutually independent behaviour).