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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