Testing whether variables are identically and independently distributed

I have the probabilities of variables that we assume initially are independently and identically distributed random variables. I know that in actuality they are not from my knowledge of the problem, but I was wondering what I can do to prove this statistically. My initial thoughts are to take a large number of samples and use the central limit theorem, but would for example testing the normality of the resulting central limit theorem be valid to disproving that they are i.i.d. If so, what would be the best test to use?

Sorry if my phrasing is wanting, I’m relatively new to this.

• The central limit theorem won't help because being iid is not necessary for convergence to a Normal, it's only sufficient. Commented Feb 16, 2022 at 5:30
• You will have to do independence tests. I figure you have continuous data, so you might want to have a look at this post: stats.stackexchange.com/questions/73646/… Commented Feb 16, 2022 at 6:14