I am a second-year undergraduate student and was reading up the topics of the law of large numbers from Casella and Berger's Statistical Inference. They state the laws as follows:
The link to the Statements of the laws as per the book
Here, As per the statements, if the same assumptions on the random variable are satisfied (The R.Vs being IIDs and EX to be finite), Then Both the laws do hold true. The book does go on to state that the condition of finite variance is not a necessary one.
It is here that I have 2 Doubts:
To what extent are the conditions accurate for either the WLLN or SLLN to be true, when stated here in the book? I somehow find it slightly difficult to believe that only the condition EX is finite can cause the laws to be valid in all possible (maybe even pathological) distributions and scenarios.
If both the Laws have the same sufficiency conditions, then how is it possible that in some cases, the WLLN may hold and the SLLN doesn't? I tried reading the Wikipedia page on Law of Large Numbers, but couldn't make sense of the examples given there with respect to what quality causes them to satisfy WLLN but not SLLN.
I hope you do see my Dilemma here. Thanks