Symmetry property of multiplicative congruential random number generator in chi-square test

The k-term prime modulus multiplicative congruential generator: Rn =(a1 Rn-1 +. + ak Rn-k) mod m, is able to produce numbers (RNs) of full period mk–1 when the multipliers al,., ak are chosen properly. In testing uniformity, the full period of RNs is usually divided into segments to calculate the chi-square statistics of the segments and test subsequently whether these statistics conform to a chi-square distribution. A symmetry property is that if an even number of segments, say 2s, is divided, then the chi-square statistic calculated from the ith segment of the first s segments is the same as that of the ith segment of the last s segments. Based on this property, the computational effort usually needed in testing uniformity is reduced by half.