I believe the most restrictive assumption we can place on a series of observations is that they are iid.
It is possible to relax these assumptions. For example relaxing the independent distribution results in independent heterogeneously distributed random variables. In other words the distribution of each random variables can itself vary. What does this mean? There must be some importance or we would not bother to specify the properties of the distribution. Can the distribution depend on preceding observations or would this break the independent observation restriction that we still have in place?
Can anyone think of any intuitive examples to help explain this (in the same way that tossing a fair coin or rolling a fair die are good examples of iid observations) and any important statistical concepts that arise out of relaxing this assumption? My degree is not in statistics although I am quite interested in this subject. Thank you all so much.