I've been exposed to many different types of convergence of random variables, namely:

  1. convergence almost surely
  2. convergence in the $r^{th}$ moment
  3. convergence in probability
  4. convergence in distribution

I feel like I have a good understanding of their definitions and I'm beginning to develop a good intuition of how they 'work' (especially with the help of some great answers on this site).

I'm new to research and I've seen papers that use these convergence theorems in different ways. I understand that a.s. convergence or convergence in the $r^{th}$ moment is preferable if possible, because they are stronger notions. But, for example, say I'm only able to show behaviour of a random variable follows some distribution of another through convergence in probability instead of almost surely, does that have an effect on my results if I'm doing some sort of statistical test?

One idea I've come across is that convergence in distribution is fine for statistical tests, but stronger convergence is necessary when trying to find estimators. Is this a correct way of thinking? Why?

I don't know if that's perfectly clear so if you have any suggestions on how to make the question more clear please let me know.