See also: pseudorandomness.
Correcting for the assumption that you are sampling from the stationary distribution of a process.
Can you infer parameters of the model if your data is gerneated by a process without a stationary distribution, or which is far from it? Related: informative sampling. Even if your data “looks” like a boring old time series, this can become a problem for long memory processes.
One solution, if your censoring process is tractable enough: hierarchical models, although you have to show that the parameter of inteest is still identifiable..
Sampling from real, time-changing problems is an issue for real data; can anyone furnish me with a large ensemble of earths to check my global economic simulations against? - so I need to know how to do it better. (See also: post-normal science)
Hurst effect. Long-memory processes. Ergodic theorems.