Relationship between probability distribution and correlation I'm unsure of the precise relationship between a probability distribution and correlation, in particular autocorrelation. What exactly is an autocorrelated probability distribution?
It seems like with a distribution we start with nothing and we can repeatedly sample from the distribution to obtain a set of values, whereas with correlation we need a pre-existing set of values to determine if there is correlation between them.
For example, say we have the uniform distribution on $(0, 1)$. We can generate a set of data by can sampling from this distribution. Now when we have the set of values we can test to see if there is any autocorrelation.
But what about if we want to generate an autocorrelated uniformly distributed, on $(0, 1)$, set of data? Does such a concept exist? It seems that if we generate autocorrelated data it isn't uniform as if it's autocorrelated it will exhibit clustering and therefore it won't be "spread evenly" between $0$ and $1$? So what exactly is an autocorrelated (uniform) probability distribution?
 A: I would like to clarify the concept behind the various names you've given, not necessarily with all the technical details, though.
A random variable (rv) is a function (with specific properties). A probability distribution is a measure on the same set on which the function (rv) operates.
An indexed series of rv's is a stochastic process (a collection of rv's) which also has (finite dimensional) probability distributions. The index is very often interpreted as "time".
Now autocorrelation is a concept which measures the correlation of values of a stochastic process at different times (cf. above). Its definition relies on the expectation and variance of the stochastic process involved (provided they exist).
So, I would say that something like an "autocorrelated probability distribution" doesn't make sense. It's the autocorrelation of a stochastic process which is usually looked at. This is not a minor detail but important to understand.
The references you've given are concerned with the "realisation" of stochastic processes having specific properties w.r.t. autocorrelation. This can be regarded as the "construction" of paths of a stochastic process with a specific distribution and autocorrelation.
Your final questions relate to the (surprisingly heavy) task of generating a series (process!) of independent uniformly distributed rv's. Here the autocorrelation plays an important role for testing independence since it is zero for independent rv's. So an autocorrelation for such a series which is different from zero indicates that it might not be the path of a stochastic process with independent rv's.
