Isn't this mainly a problem with linear congruential generators (especially the low bits of a power-of-two-modulus LCG), rather than all PRNGs? If you want good-quality random numbers, read from /dev/urandom. It's also quite fast -- even on my low-end 8-year-old laptop, it can produce over 10^8 random bytes per second: $ time bash -c 'head -c 500000000 /dev/urandom > /dev/null' real 0m3.901s user 0m0.070s sys 0m3.815s Best wishes, Adam P. Goucher
Sent: Sunday, March 10, 2019 at 10:31 PM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: "Dan Asimov" <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Statistics for a class of random solids
Where higher dimensional geometry is involved, bear in mind that even a well-designed pseudo-random number generator is only independent to some fixed precision. Once dimension x (user-demanded precision) exceeds this quantity, the randomness becomes compromised: heed the awful warning in G. Marsaglia "Random numbers fall mainly in the planes" https://www.ncbi.nlm.nih.gov/pmc/articles/PMC285899/pdf/pnas00123-0038.pdf
This effective precision is invariably omitted from a PRNG specification, and must instead be deduced from a detailed inspection of the algorithm.
WFL
On 3/10/19, Dan Asimov <dasimov@earthlink.net> wrote:
Wait — you mean that isn't the actual origin of the name "Box-Muller" ?
—Dan
----- Note that the probability density function on R^2 where each coordinate is independently uniform [0, 1] is shaped like a box. As such, it seems reasonable to refer to the shape of the probability density function of a bivariate standard Gaussian as a 'muller', so that the Box-Muller transform lives up to its name. -----
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