The stream cipher RC4 (also known as ARCFOUR) exhibits a very small bias towards succesive characters from the CPRNG being identical: this probability is not 2^-8 but closer to 2^-8 + 2^-24. This has been experimentally verified with just under 56 trillion samples of RC4 output.
Note that the work here is superceded by this paper:
Statistical Analysis of the Alleged RC4 Stream Cipher, Scott R. Fluhrer and David A. McGrew, Fast Software Encryption Seventh International Workshop, Springer-Verlag, March, 2000.
[ PDF ] [ David McGrew's home page ]
You can fetch the source of the program used to do the tests. I've also made available a complete archive (gzipped tar file, 621k) of the program and the results generated, which includes a Perl script for reading the results. The script generates the output below. Note that the probability of a result being 7.394 SDs or more from the mean of a normal distribution is less than 1.5 * 10^-13.
Number of tests: 256 * 218634000000 - 15199 Number of coincidences: 218634000000 + 3450619 Number of timed tests: 217914000000 Time in seconds for timed tests: 8747636 Timed tests took 8747636 s total, 6377263.98309212 tests/sec 2 ^ 45.6697268168272 tests SD 2 ^ 18.832040126843, SD of sample measurement 2 ^ -26.8376866899841 95% confidence interval 2 ^ -25.8668330356437 Trying hypothesis: p(coincidence) = 1/(2^8) Delta is positive 2 ^ 21.7183989336417 difference, proportion 2 ^ -23.9513278831854 Distance in SDs 7.39401928729239 Trying hypothesis: p(coincidence) = 1/(2^8) + 1/(2^24) Delta is positive 2 ^ 16.8046042398197 difference, proportion 2 ^ -28.8651225770074 Distance in SDs 0.245290644157141