# Difference in conditions of Weak and Strong Law of Large Numbers

I understood the difference between convergence in probability and almost surely. And after searching online, I found multiple counterexamples to show that the WLLN does not necessarily imply the SLLN. However I am still failing to find the difference in conditions between the two as can be seen here.

The conditions, as I can see, are: independent and identically distributed random variables, with finite mean $\mu$ for both of them. Thus if conditions of one law are satisfied, the conditions of the other are also satisfied and vice versa.

What am I missing? I am not looking for counterexamples, as I have found countless of those. Thanks in advance.

• If I understand your question correctly, I think that what you are missing is that when one says WLLN $\not\Rightarrow$ SLLN it means that the conclusion of the WLLN does not imply the conclusion of the SLLN. Or, equivalently, the WLLN is not an equivalence. Commented Jun 3, 2018 at 15:57
• Say I have a sequence random variables...and I want to check whether the WLLN and/or the SLLN holds. This means I have to check if the conditions mentioned (i.i.d and finite mean) are satisfied. Let's say they are satisfied...then I would obviously conclude that BOTH of them hold. In which case would the WLLN be satisfied and not the SLLN? Commented Jun 3, 2018 at 16:02
• "Check whether the WLLN holds" does not mean check that the conditions to apply it hold, but check that the conclusion holds. I.e., check that there is convergence in probability. Commented Jun 3, 2018 at 16:03
• And how would one check for the conclusion without checking the conditions that would lead to it? Commented Jun 3, 2018 at 16:04
• In a pedestrian manner. Now this is just my interpretation of your question, you may get better points of view from people more expert than I am. Commented Jun 3, 2018 at 16:11