This constitutes one of the well known tests of randomness applied to pseudo-random number generators - the so-called spectral test, the object of which is to discover the behavior of a generator when its ouputs are used to form n-tuples. The formal development of this family of tests is difficult and we will not cover it here (the interested reader is referred to Knuth), but we will attempt to illustrate through several examples the meaning of the concept as it applies to a 2-dimensional setting and with linear congruential generators. The concepts presented are valid in higher dimensions as well.
Figure 6(a): Pair Plot of LCG(10,0,509,1).
Figure 6(b): Pair Plot of LCG(128,0,509,1).
Figure 6(c): Pair Plot of LCG(108,0,509,1).
Figure 6(d): Pair Plot of LCG(269,0,2048,1).
Figure 6(e): Pair Plot of LCG(1203,0,2048,1).
Figure 6(f): Pair Plot of LCG(16807,0,2147483647,1).