# Is it correct that no statistical simulation is done using $2^{256}$ (or more) outputs from a MWC prng?

In a conversation about Marsaglia's Multiply-With-Carry pseudo-random number generator and its potential use in generating random data for statistical purposes, someone said:

There is no statistical simulation ever done that will use $2^{256}$ outputs from such a generator.

Is that correct?

• If "yes" — Is there an explanation which might help me understand why that is correct? And is there any other statistical purpose where more than $2^{256}$ of a MWC prng output would be used?
• If "no" — Can you give me a name of a statistical simulation I could mention as an example?

The reason why I'm asking is because (being a stats-noob) I trusted in the fact that Marsaglia (being a stats-guru) had a good reason to develop the MWC prng with its "near-record period". But now, I have a hard time understanding for what statistical purposes the MWC prng might come handy if it's indeed true that no statistical simulation will ever use $2^{256}$ (or more) outputs from such a generator.

It's a bit hard to say 'no statistical simulation will ever', since forever is a very long time and we may find ways to do things we can't see any way to do now.

However, for the foreseeable future, $2^{256}$ or $\sim 10^{77}$ simulations is so many orders of magnitude beyond what we'd be able to generate in reasonable time with current understanding of computation (let alone what we could ever really need for statistical purposes, which may well be even less) that I doubt it will even come remotely close before MWC is a long-forgotten footnote in the annals of random number generation. It may be difficult to appreciate quite how large that quantity is.

Imagine you had $7\times 10^{18}$ cores (a billion cores for each and every person on the planet), each screaming along at a billion simulations a second. Let it run for a century (about $\pi$ billion seconds). That would be a simulation of size roughly $10^{-40}$ times $2^{256}$. You'd have to let it run for $10^{40}$ centuries

(The earth has been around for a bit over $4\times 10^7$ centuries)

So not any time soon.

(And if we suppose technologies like quantum computers... why would we use MWC?)

Practically speaking, I think the person you quote is correct. Long before the time we get anywhere near that issue, we would have ceased using MWC.

(My answer deliberately avoids addressing any issues of the suitability of MWC in general, since it's not directly relevant to the question, though would be a consideration if one comes to use it for something.)

As to why anyone would develop a generator with a period far longer than you might need for any one simulation; one advantage is that you don't have to worry about dependence between two simulations with different seed - if I do a simulation today with one seed and another tomorrow with a second seed, the sets of random numbers I get will almost never have any overlap.

• Thanks for the quick answer (and the extended edit). Much appreciated. [+1] – e-sushi Sep 25 '13 at 2:32
• You might actually add an energy analysis as well. – orlp Sep 25 '13 at 9:31
• @nightcraker quite – Glen_b Sep 25 '13 at 10:22
• Vivid examples, cheers. – conjectures Nov 10 '13 at 10:45