I would like to apply a permutation test to a sequence with 4,000,000 elements. To my knowledge, it is infeasible due to a number of possible permutations being ridiculously large (no RNG will generate uniformly distributed values in range `{1 ... 4000000!}`). I've heard of pseudorandom permutations though, and it sounds like something I need, but I can't comprehend if it's actually a proper replacement for random shuffle in my case.

If you are running a permutation test I presume that you want to generate a random sample from the set of all possible permutations, so that you can test some statistic calculated on the real data against the distribution of statistics calculated on the permuted data.

Algorithms for generating random permutations, such as those described at http://en.wikipedia.org/wiki/Random_permutation, typically use many random numbers, so there is no requirement for any single step of the generation process to need numbers as large as 4000000!. The only worry would be that, since the seed used to generate the random numbers is typically much smaller than 4000000!, not all permutations are possible.

There are other statistical tests which consume very large quantities of pseudo-random numbers (e.g. MCMC), so I wouldn't worry about this if you are using a random number generator which is commonly used for statistical tests. If you are worried about this, you could repeat the test with a cryptographically secure random number generator, such as http://docs.oracle.com/javase/6/docs/api/java/security/SecureRandom.html. This will be slower, so you might need to reduce the number of permutations tested, but it is very unlikely that it has any characteristic which would stand out far enough to affect your test results, because any such characteristic would be a security weakness - it would mean that, given a large quantity of random numbers already generated, you would have a slightly better than random chance of guessing the next number correctly.

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