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.


Your tests are events or random variables defined on the probability space associated with the null hypothesis. Some tests are independent, while some are not.

The particular counts (on a finite subsequence) are not independent random variables since they attain their maximums at the same all-1 sequence whose probability isn't equal to its square.


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