title: Is there a way to test the quality of a PRNG for multidimensional use?
body: <p>I'm in the process of evaluating some PRNGs, both in terms of speed and quality. One aspect of quality I want to test is multidimensional distribution and bias.</p>
  
  <p>I know of <a href="http://en.wikipedia.org/wiki/TestU01" rel="nofollow">TestU01</a>'s batteries, and I plan on using them (and, perhaps, others that the <a href="http://csrc.nist.gov/groups/ST/toolkit/rng/batteries_stats_test.html" rel="nofollow">NIST</a> suggests).</p>
  
  <p>But what about testing multidimensional bias? <a href="http://www.boost.org/doc/libs/1_56_0/doc/html/boost_random/reference.html#boost_random.reference.generators" rel="nofollow">Boost's PRNG</a> have some comments, and the Mersenne Twister is known to be uniform in several hundred dimensions, while the Hellekalek PRNG has good uniform distribution in "several" dimensions (however many that means...).</p>
  
  <p>I imagine the runtime complexity of a battery testing for multidimensional bias would increase with each dimension. So it's possible there isn't a suitable battery for this test. However, that I haven't confirmed that suspicion.</p>
  
  <p>Is there a known way to test PRNGs for multidimensional bias? I'd even be okay if the test is limited to 2, 3, or 4 dimensions; that would be better than no test at all.</p>