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I am a little confused by what conditions need to hold for the weak law of large numbers (WLLN) and the strong law of large numbers (SLLN) to be true. It seems different sources give me different answers, and I am not sure if this is a result of people recently managing to prove the WLLN/SLLN with weaker conditions, or just my misunderstanding.

So, if we wanted $X_1,...,X_n$ to converge to $\mathbb{E}[X]=\mu$ almost surely, in the statement of the SLLN in Casella and Berger, they say we need that $\mu$ is finite and $\mathbb{E}[X^2]$ is finite. They go on to say that this is actually stronger than necessary, the SLLN only needs $\mathbb{E}[|X|]$ to be finite (this doesn't imply finite second moment, and hence doesn't imply finite variance). They also say the WLLN also doesn't need finite second moment, and only $\mathbb{E}[|X|]$ to be finite. I have also seen text books which say that the SLLN requires finite fourth moment! It seems that textbooks can't come to a consensus!

So, what actually are the conditions for WLLN and SLLN in terms of the moments of the distribution? Am I right in saying that the only difference between the SLLN and the WLLN is that the WLLN doesn't require finite expectation, whereas the SLLN does?

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    $\begingroup$ SLLN = Strong Law of Large Numbers, WLLN = Weak Law of Large Numbers $\endgroup$
    – R Carnell
    Commented Dec 31, 2020 at 19:31
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    $\begingroup$ Wikipedia has some examples where the WLLN holds but the SLLN doesn't: en.wikipedia.org/wiki/… $\endgroup$
    – fblundun
    Commented Jan 2, 2021 at 13:32

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I think that Casella and Berger are choosing their conditions to match the narrative of the chapter. They are covering Convergence Concepts in that chapter, and so they moving through Convergence in Probability, Consistency, the weak law of large numbers (WLLN), the central limit theorem, almost sure convergence, the strong law of large number (SLLN), etc. For the WLLN, they choose to use Chebychev's Inequality for a proof, and so they require a finite second moment. For the SLLN, they keep the finite second moment, but say "The only moment condition needed is that $E(|X|) < \infty$". The proof is likely more difficult than they want to present, which is why they give a reference. For your other citations where stronger conditions are required, it might be based on the proof that they are choosing to present in their text. Clearly the Cauchy distribution is the one you want to concentrate on to relax the second moment requirement.

For your final question, the difference between the SLLN and WLLN is the strength of the convergence statement (probability of a limit versus the limit of a probability).

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    $\begingroup$ I understand the consequence of the theorems are different, but there is also a difference needed for the conditions of each theorem, else the weak law of large numbers would be pointless (we would always just use the SLLN if the conditions were the same). My question is, if all we require in both theorems is that $E[|X|]$ is finite, why would we ever use WLLN? $\endgroup$
    – jacob
    Commented Dec 31, 2020 at 20:31

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