Task! Desired statement: Let $\seq{X_n} \subset L^1$ be a family of independent, identically distributed random variables on the same probability space with $\ev(X_1) = 0$ and $\norm{X_1}_2 = 1$. Let $S_n = \sum_{k \le n}X_n$ and $\Gamma_n = \sqrt{2n\ln(\ln(n))}$ with $n \ge 3$. Then $ \limsup \frac{S_n}{\Lambda_n} = 1 \quad \liminf \frac{S_n}{\Lambda_n} = -1 $ Additional assumptions allowed: $\seq{X_n} \subset L^4$. > The strong law of large number is the asymptotic under the conservation of the first moment. CLT is the asymptotic under the conservation of the second moment. Try 0-1 law into pushing computations outside.