K-dimensional equidistribution - how big are periods of PCG? I'm trying to figure out what is period of PCG generator XSL-RR-RR:
https://www.pcg-random.org
https://en.wikipedia.org/wiki/Permuted_congruential_generator
If we use just random multiplier and increment in LCG in that PCG. M. O'Neill wrote:
&quot;For the PCG family, arbitrary k-dimensional equidistribution (and the huge periods it implies) requires PCG's extended generation scheme.&quot;
It looks like their generators has big periods, XSL-RR-RR too. But I can't find accurate information about that. She proved somehow that they are k-dimensional equidistributed. But what is k exactly, what periods it implies and could it be the same with every parameter of LCG in XSL-RR-RR? She tested some specific multiplier and increments so maybe other ones have worse parameters and periods?
I don't understand exactly why she use just some specific miltipliers (two ones if I understand it well) and increments in XSL-RR-RR. What will happen if we will use other, random ones?
PCG is just LCG where the output is run through a transformation. The amount of internal state of the PCG - which is what controls the period - is all contained in the underlying LCG. So the period is $2^n$ where $n$ is the number of bits of state. The only thing to worry about is if the output transformation makes the period smaller, but since it's a permutation it does not.
When pcg-random.com talks about &quot;arbitrary&quot; period, they just mean that they have a way to make the internal state arbitrarily large. For a given state size, the period is the same as other algorithms of the same size.