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.

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.
