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Statistical independence properties of pseudorandom vectors produced by matrix generators. (English) Zbl 0708.65007

For a short general discussion of the generators see p. 87 of [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).
Reviewer: K.G.Brokate

MSC:

65C10 Random number generation in numerical analysis
11K45 Pseudo-random numbers; Monte Carlo methods
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