A nonstandard random number generator?

2013 March 6
by Daniel Lakeland

I’m throwing the following idea out for consumption, and see if anyone has anything useful to say about it.

Suppose we’re working in the IST form of nonstandard analysis. Suppose that we have a sequence of irrational real numbers

$$(\xi_i), i \in \mathbb N$$ such that whenever $$N$$ is a nonstandard natural number and $$n$$ is a standard natural number

$$\frac{1}{N}\sum\limits_{i=1}^N I(dV_n,\xi_{(i+k) \times (i+k+1) \times … (i+k+n)}) \cong dV_n\, \forall dV_n \subset [0,1]^n , k \in \mathbb N$$. Here $$I$$ is the indicator function for the volume $$ dV$$. I suppose that we should allow $$dV$$ to be infinitesimal, and put a requirement on $$N$$ related to the size of $$dV$$, perhaps $$N > 1/dV^2$$. This only matters for infinitesimal volumes.

Roughly, for any standard number of dimensions, a sample of vectors of length n, of nonstandard size will have a number of points inside a volume $$dV$$ proportional to the volume, regardless of the starting point in the sequence $$k$$.

This sequence is basically a model for an ideal random number generator of the type used in computers. It’s not clear to me whether such a sequence can be shown to exist. Perhaps an infinite number of such sequences exists! But if it could, this would seem to be a nice basis for matching probability theory to probability practice.

 

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