Is it possible to generate data for stochastic process with specific distribution and autocorrelation? It seems though that there is a disconnect between constructing paths of a stochastic process with both a specific distribution and autocorrelation. It seems like you can have either one property or the other, but not both?
For example, say at time $t_1$ we sample from a uniform distribution and obtain some value $x_1$. Now at the next time step $t_2$ we again sample from the uniform distribution and obtain another value $x_2$. And so on...
These values are chosen at random on each time step so they are surely uncorrelated? However if we generated them such that they were autocorrelated then surely we aren't randomly sampling from a uniform distribution at each step as the autocorrelation will mean that some values are more likely to be chosen than others? So how can we have a stochastic process with both a specific distribution and autocorrelation?
 A: I would like to refer to my answer on your previous question (Relationship between probability distribution and correlation). In a nutshell, what you asked for can be achieved but it is (surprisingly) hard to do so.
Without going into the details here, what you could look for are techniques called (V)ARTA/NORTA which reads "(vector) autoregressive/normal to anything". These allow for the simulation of rv's and stationary stochastic processes with a prescribed probability distribution and correlation structure.
For ARTA, which should be the interesting one here, a reference is
M. Cario and B. Nelson. Autoregressive to anything: Time-series input processes for simulation. Operations Research Letters, 19:51–58, 1996.
But beware, fitting the simulated process path in a way that it exhibits a desired autocorrelation is a hard task and involves higher computational effort, at least then using the techniques above.
Of course there are other possible approaches (e.g. copulas).
A: I don't have a lot of experience with generating autocorrelated processes, but I'd probably start my exploration here:
The problem you describe is that each time step, you sample from a separate (uncorrelated) distribution. The solution then should be to sample from a multivariate distribution with an appropriate covariance matrix. 
The covariance matrix should be designed such that the variables in the distribution are autocorrelated.
As a simple example, imagine you have 4 time periods, and you want to generate some autocorrelated process. Assume a multivariate Gaussian $N((\mu_1, \mu_2, \mu_3, \mu_4), \Sigma)$, where $\Sigma$ is the covariance matrix.
The diagonal elements of a covariance matrix are just variances of the four random variables. But the off-diagonals are covariances. So you can design the covariance matrix such that the elements are correlated over time.
For example, you could have:
$ \Sigma = \begin{smallmatrix} 4.  &  3.6 &  2.  &  0.4 \\
3.6 &  4. &  3.4 &  2.6 \\
2.  &  3.4 &  4. &  3.48 \\
0.4 &  2.6 &  3.48 &  4. \end{smallmatrix} $ which will make the variables (auto)correlated. But also stochastic (randomly drawn from a multivariate distribution).
Am I inching closer to what you were looking for?
A: Keeping things simple:
Would it be possible to model directly the conditional probability $U(X_{t+1} | X_{t}, X_{t-1}, ..)$ ? Perhaps assuming that the process is Markovian, i.e. $U(X_{t+1} | X_{t})$? Here $t$ is a time index: $t+1$ is the next step..
If we're having a discrete process, uniform distribution and under the assumption of Markovianity, that should be "feasible"..
