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Thus, the theorem states that sample averages approximate the value of an integral. It is to be noted that the only property of the sequence (xj)i°° employed is its equidistribution. In applications, one may employ several equidistributed sequences simultaneously; accordingly, new requirements may arise. The sequences may be employed, for example, as bases for decision, in which case it may be required that they be independent. Thus depending on the Monte-Carlo problem considered, equidistributed sequences may be required to possess other random number characteristics. Let the sequences (x3)i°°, (xj+r)i°° be designated respectively by x, x(t), in which t is a non-zero integer, then an additional desirable characteristic is the statistical independence of x, xM. A sequence exhibiting such characteristics is ({aj })ix in which a is an irrational number [2]. The symbol jx} is employed to designate the fractional part of x. However, from the viewpoint of the practical utilization of the sequence suggested above by means of a digital computer, it is necessary to replace a by a rational number, and hence, to lose some of the precision with which the characteristics discussed above are satisfied. Accordingly, the sequences which will be studied are

The autocorrelation and joint distribution functions of the sequences {𝑎\over𝑚𝑗²}, {𝑎\over𝑚(𝑗+𝜏)²}