Suppose I have a sequence $\{x_i\}_{i=1}^\infty$ of zeros and ones. I want to test if they are randomly generated according to a conjectured scheme (the example to keep in mind is that they are conjectured to be generated by independent coin flips). I can use a statistical test --say to see if the sequence 11 occurs as much as it should up to some bound $x$, or I can use a different test --say to see if $1r1$, where the $r$ is anything--occurs as much as it should up to the same bound (in general I'll be interested as $x \rightarrow \infty$).

The question is, does knowing that one test suggests that this is a randomly generated set guarantee (or somehow influence) the other test's results? Are they strongly correlated? Are there tests which would be correlated? I believe this study falls under the subject of 0-1 laws, but do not know where to begin looking. Any references, books, papers, answers would be appreciated--I'm still trying to find my way around the field.

Also please retag if you know of better tags to use.