Timeline for Generalization of Law of Large Numbers and Central Limit Theorem using Polynomial Scaling

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Dec 5, 2016 at 0:06	comment	added	John Doe		(2) I didn't understand why $(R_n Y_n)_{n \geq 1}$ does not converge and why does it imply that $S_n^a$ doesn't converge for any $a \in (1, 1/2)$, could you give me some hints please.
Dec 5, 2016 at 0:06	comment	added	John Doe		But I have two follow up questions. (1) By law of iterated logarithms I understand that $S_n^{1/2}/\sqrt{2 \log \log n}$ finite a.s, but how does it imply (along with the "in probability" convergence proved by you) that $S^a_n = 0$ a.s. for $a > 1/2$?
Dec 5, 2016 at 0:05	comment	added	John Doe		Thank you for the answer, @Davide Giraudo. I looked at Marcinkiewicz law of large numbers, it gives exactly what we want for $a \in (1/2, \infty)$. Just to make it clear I understand by Slutsky's lemma we can say that $S^a_n \overset{d}{\to} 0$ and convergence in probability follows from the fact that $X_n \overset{d}{\to} c \implies X_n \overset{p}{\to} c$, where $c$ is a constant.
Dec 2, 2016 at 10:07	history	edited	Davide Giraudo	CC BY-SA 3.0	added 67 characters in body
Nov 30, 2016 at 22:08	history	answered	Davide Giraudo	CC BY-SA 3.0