Necessary moment condition for the law of the iterated logarithm

Question

Let $X_1, X_2,\ldots$ be iid random variables, and let $S_n = X_1 + \cdots + X_n$, $n \geq 1$. It seems classical that if $$ \limsup_{n \to \infty} \frac {|S_n|}{\sqrt {2 n \ln \ln n}} < \infty $$ almost surely, then $E(X_1^2) < \infty$ (and $E(X_1) = 0$). But what is a proof of this result?

Answer 1

This is a theorem due to Strassen, whose proof is based on Skorokhod's representation of zero-mean distributions on $\mathbb R$ by means of the Brownian motion.

Heyde gave a direct proof of this result, without using Skorokhod's representation.

This result is cited e.g. in Supplement 1 in Section 4 of Chapter X of the book by Petrov.