2
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ Do you have a response to the answer below? $\endgroup$ Oct 20, 2021 at 21:10

1 Answer 1

4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.