Suppose $X_i$'s are non-degenerate i.i.d. Then
(1) If $E|X_1|^p=\infty$ we've $\limsup_{n\rightarrow \infty} \frac{|S_n|}{n^{1/p}}=\infty$. And this is true $\forall p>0$
(2) However for $p=2$ we can say more. For every choice of centring constant $\{C_n\}$ we have $\limsup_{n\rightarrow \infty} \frac{|S_n-C_n|}{\sqrt{n}}=\infty$. i.e. Does not matter whether $E(X_1^2)<\infty$ or not!
I know the proof but whenever I look at them they does not stop surprising me as if something's wrong in there(in (2)). Is there any way to visualize what's going on here? Something related to laws of iterated logarithm (where we assumed finite $2^{nd}$ moments)? Would you please shed some light on it or/and suggest some reading materiel? Thank you,