To test a 'random' bit sequence for cryptographic strength I have been using the NIST's Non-overlapping Template Matching Test (See page 2-14 but I will explain here). This test uses a non-periodic bit template $B$ of length $m$ and counts the occurrences of the template in the bit sequence over $N$ blocks (where the blocks are just the original bit sequence split up into chunks). Then a Chi-squared test is done.
Let's say $B$ is $0001$, which is non-periodic.
If the bit sequence is $000010100010000...$ then we have...
000010100010000... 0001 no match 0001 match. advance by m 0001 no match 0001 no match 0001 match. advance by m... 0001 no match. ...
Now the question is why should we particularly care about the distribution of non-periodic patterns as compared with periodic ones? Are they special in some way as related to random number generators?