How to determine if binomial events are independent? I have a sequence of binary experiment results, something like 1100010000100...
My first hypothesis is that these events are independent, but I'd like to know if there is some way to test this.  I could look at the probability of a 1 immediately following another 1, for example, and see if it is close to the overall probability of a 1, but that's just one possible kind of correlation/dependence.  Is there some more general way to look for patterns?  
Two things that seem like they might be helpful are Fourier transforms and hidden markov models, but I don't really know enough about either to say whether they apply to this situation.  Even just some pointers for further reading would be very helpful.
 A: The difficulty is that "more general" covers an enormous number of possible forms of dependence (potentially infinitely many).
If you specify a form of dependence, then it's easy to test. You mention specifically considering the dependence between successive outcomes (by considering the comparison between the conditional $P(X_t=1|X_{t-1}=1)$ and the marginal $P(X_t=1)$. That kind of serial dependence would be the most obvious single test to consider.
But what if $P(X_t=1)$ depends on some non-obvious function of $X_{t-2}$, $X_{t-5}$, $X_{t-13}$, $X_{t-38}$, and $X_{t-174}$ say?
Many tests of dependence have been proposed at various times. André Nicolas mentioned the runs test, for example. There are a variety of runs tests, but in the cases of 0/1 data the obvious run to consider is 'runs of one kind' -- the distribution of the lengths of runs of 1's, 0's (or potentially, both). That tends to pick up the kind of dependence that induces longer- (or shorter-) than-expected runs for independent data, for example.
