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.]*

You wrote: “The concepts of convergence in probability and almost sure convergence in probability theory are specialisations of the concepts of convergence in measure and pointwise convergence almost everywhere in measure theory.”

In measure theory, is there any different between “pointwise convergence almost everywhere” and “convergence almost everywhere”?

*[No – T.]*