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up vote 0 down vote favorite Let $S_n sim N(0,n)$ so that $S_n$ is the partial sum process of standard normal random variables. My objective is to prove: $$sum_{n=1}^infty P(S_n geq (1+epsilon)sqrt{2 n log log n }) < infty$$ for all $epsilon > 0$ . I cannot use the law of iterated logarithm, as the reason I want to prove the above inequality is to prove a weaker form of iterated logarithm ( $leq 1$ instead of $=1$ ). My attempt has been to bound the tail probabilities using known inequalities for standard normal. Firstly, we rewrite it as $$sum_{n=1}^infty P(S_n/sqrt{n} geq (1+epsilon)sqrt{2 log log n }) < infty$$ $$sum_{n=1}^infty (1-Phi((1+epsilon)sqrt{2 log log n })) < infty$$ Inequalities I know of: $$1-Phi(x) leq x^{-1} e^{-x^2/2}/sqrt{2pi}$$ $$1-Phi(x) leq frac{1}{2} e^{-x^2/2}$$ But neithe