Imagine the following experiment: someone is sitting behind the screen and calls out a sequence of numbers: "1! 3! 5! 3! 4! ...". Let's say he/she and I agreed beforehand that all numbers are integers from 1 to 6. Can I find out if to generate these numbers he/she is using a truly random process (like throwing an ideal die or performing quantum-mechanical experiment) or he/she just has a sophisticated pseudo-random algorithm?
Note, that we haven't agreed about a particular probability distribution, so it is up to my opponent to choose it. Therefore I can't use $\chi^2$-test (or can I?).
A related question, because I expect the answer to the first one is "No". Imagine the same setup as above, but now it is me who chooses the probability distribution. Are there any distribution such that my opponent wouldn't be able to deduce a pseudo-random algorithm to mimic it. Or it will be very hard (slow convergence)?
See e.g. Cryptographically secure pseudorandom number generator. A high-quality pseudorandom generator will be extremely difficult to distinguish from random. However, in principle, if you are lucky enough to guess which algorithm your opponent is using (including the size of the seed), and have a lot of time on your hands, you could try all possible seeds, and after observing enough output you could tell which seed is being used and then be able to predict with certainty the output from then on.