I am trying (for the sake of my own better understanding) to replace all of the big-O’s with explicit constants. I am having trouble with display (9), however. Is it being claimed with high probability for all n, for n large enough, almost surely for all n, etc?

Best regards,

-Aryeh

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.

]]>