Thanks for a great post!

Could you please explain the last claim of remark 2 in more detail? In Counterexamples in Probability by Romano & Siegel (pg. 113), there is a nice example of a triangular array of IID random variables that have moments of order p when p 2. But the p = 2 case is trickier and I could come up with neither proof nor counterexample. What did you mean when you said the Chernoff bounds work for the p = 2 case as well?

Much thanks!

]]>So if we tried to get around requiring , the strong law of large numbers, as stated, would fail.

]]>One thing that’s been bothering me for a while: why is the absolute value sign used in the statement of the theorems? Why do we need E(|X|) instead of just E(X)?

Thank you.

]]>*[Corrected, thanks – T.]*