Marsaglia's theorem

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The set of Marsaglia (by George Marsaglia , 1968 ) states for a linear congruential (qv) gained pseudorandom numbers following: ${\displaystyle y_{i+1}\equiv (ay_{i}+r)\,{\bmod {\,}}m}$ ${\displaystyle u_{i}={\frac {y_{i}}{m}}}$

Is formed from the result of the tuple , ..., ..., so by this are certain tuple points in a maximum of parallel hyperplanes . ${\displaystyle (u_{i})_{i\geq 0}}$ ${\displaystyle (u_{0},...,u_{k-1})}$ ${\displaystyle (u_{1},...,u_{k})}$ ${\displaystyle (u_{n+1},...,u_{n+k})}$ ${\displaystyle k}$ ${\displaystyle \mathbb {R} ^{k}}$ ${\displaystyle {\sqrt[{k}]{m\cdot k!}}}$

The following components are used here:

• Module , ${\displaystyle m\in \mathbb {N} }$
• Increment , ${\displaystyle r\in \mathbb {N} _{0}}$
• Factor , ${\displaystyle a\in \mathbb {Z} }$
• Start value (seed) . ${\displaystyle y_{0}\in \left\{0,...,m-1\right\}}$

The set is therefore suitable for testing linear congruence generators, since with a higher number of parallel hyperplanes, a higher quality of the pseudo-random numbers can be assumed.

The inverse congruence generator does not know such hyperplane behavior and is therefore an alternative to the linear congruence generator.

Marsaglia's theorem is used in the spectral test for testing random numbers.