discrete stochastic process: exponentially correlated Bernoulli? 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?)
 A: In other words:
Start with a random variable $X_0$ Bernoulli with parameter $p$, random variables $Y_n$ Bernoulli with parameter $\alpha$, random variables $Z_n$ Bernoulli with parameter $p$, and assume that all these are independent. Define recursively the sequence $(X_n)_{n\ge0}$ by setting $X_{n+1}=Y_nX_n+(1-Y_n)Z_n$ for every $n\ge0$.
Then $X_n$ and $X_{n+k}$ are conditionally correlated if and only if $Y_i=1$ for every $i$ from $n$ to $n+k-1$. This happens with probability $\alpha^k$, hence you are done.
This is Douglas Zare's idea, but with no Poisson process.
A: I suggest also to look a the paper:
Generating spike-trains with specified correlations.
By Jakob Macke, Philipp Berens, et al. (Max Planck Institute for Biological Cybernetics.).
Generating spike-trains with specified correlations
They also offer a Matlab Package for 'Sampling from multivariate correlated binary and poisson random variables' ... also available at Matlab central:
Sampling from multivariate correlated binary and poisson random variables
Also look at the page 
link
A: The above solution is very nice, but relies on the very special structure of the desired process. In a much more general framework, I think that one could use a perfect simulation algorithm as described in:
Processes with long memory: Regenerative construction and perfect simulation,
Francis Comets, Roberto Fernández, and Pablo A. Ferrari,
Ann. Appl. Probab. 12, Number 3 (2002), 921-943.
A: Here is a construction.


*

*Let $\{Y_i\}$ be independent Bernouilli random variables with probability $p$. 

*Let $N(t)$ be a Poisson process chosen so that $P(N(1)=0)=\alpha$.

*Let $X_i = Y_{N(i)}$.


In words, we have some radioactive decay which tells us when to flip a new (biased) coin. $X_n$ is the last coin flipped at time $n$. The correlation between $X_m$ and $X_n$ comes from the possibility that there are no decays between time $m$ and time $n$, which happens with probability $\alpha^{|m-n|}$. 
The conditional correlation between $X_m$ and $x_n$ is $1$ if $N(m) = N(n)$, and $0$ if $N(m)\ne N(n)$, so  $\text{Cor}(X_n,X_m) = P(N(m)=N(n)) = \alpha^{|m-n|}.$
You can simplify this by saying that $N(i) = \sum_{t=1}^i B_i$ where $\{B_i\}$ are independent Bernoulli random variables which are $0$ with probability $\alpha$.
