The Other Law of the Iterated Logarithm
The Other Law of the Iterated Logarithm
Let $\{X_n\}$ be a sequence of independent, identically distributed random variables with $EX_1 = 0, EX_1^2 = 1$. Define $S_n = X_1 + \cdots + X_n$, and $A_n = \max_{1\leqq k\leqq n} |S_k|$. We prove that $\lim \inf A_n(n/\log \log n)^{-\frac{1}{2}} = \pi/8^{\frac{1}{2}}$ with probability one. This result was proved …