Are there any practical applications of the Strong Law of Large Numbers for which the Weak Law of Large Numbers would not suffice?
When, in practice, is the result $$\lim_{n\rightarrow\infty} \mathbf{P}\biggl[\Bigl|\frac{1}{n}\bigl(X_1+X_2+\cdots+X_n\bigr)-\mu\Bigr|\ge\varepsilon \biggr] = 0$$ not enough?
When is $$\mathbf{P}\biggr[\lim_{n\rightarrow\infty}\frac{1}{n}\bigl(X_1+X_2+\cdots+X_n\bigr) = \mu\biggl]=1$$ needed?
Thank you!