There is a question that was asked on stackoverflow that at first sounds simple but I think it's a lot harder than it sounds.

Suppose we have a stationary random process that generates a sequence of random variables x[i] where each *individual* random variable has a Bernoulli distribution with probability p, but the correlation between any two of the random variables x[m] and x[n] is α^{|m-n|}.

How is it possible to generate such a process? The textbook examples of a Bernoulli process (right distribution, but independent variables) and a discrete-time IID Gaussian process passed through a low-pass filter (right correlation, but wrong distribution) are very simple by themselves, but cannot be combined in this way... can they? Or am I missing something obvious? If you take a Bernoulli process and pass it through a low-pass filter, you no longer have a discrete-valued process.

(I can't create tags, so please retag as appropriate... stochastic-process?)