# Can the Law of the Iterated Logarithm be strengthened?

http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm

.$\quad$1. Can the independence assumption be weakened, similar to this?

.$\quad$2. Can the identically distributed assumption be dropped/weakened, in the latter case similar to this?

.$\quad$3. Can the result be fine-tuned, presumably to something of the form

$\displaystyle\limsup_{n\to \infty} \frac{\frac{S_n}{\sqrt{2\cdot n\cdot \operatorname{log}(\operatorname{log}(n))}}-1}{f(n)} \; \;$ some_relation_symbol $\;$ some_constant $\qquad$ almost surely $\qquad \; \;$ ?

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For your 2nd question take a look at Theorem 34.2 (of Kolmogorov) in the book "Probability Theory" by Heinz Bauer. I got the correct page in Google books by googling for bauer kolmogorov petrov stout "probability theory". –  j.p. Jul 19 '11 at 9:43
For the 3rd question, but still in the IID case. In a certain sense the refinement is the Berry-Esseen Theorem. en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem –  Gerald Edgar Jul 19 '11 at 14:19
Berry-Esseen is completely different. BE quantifies the rate of convergence in the Central Limit Theorem, LIL quantifies the rate of convergence in the Strong Law of Large Numbers. –  Ricky Demer Jul 19 '11 at 18:02

For your third question, see a paper of Erdös, where he proves an even more precise result (at least, in the special case $S_n=\sum_{i=1}^nY_i$ where $Y_i$ (independent) are $\pm 1$ with probability $\frac 12$), namely that for $\delta>0$, the following holds with probability one: $$S_n>\left(\frac{2n}{\log\log n}\right)^{1/2}(\log\log n+\frac 34\log\log\log n+\frac 12\log\log\log\log n+\cdots+(\frac 12-\delta)\log^{(k)}n)\qquad\text{for infinitely many n}$$ and $$S_n>\left(\frac{2n}{\log\log n}\right)^{1/2}(\log\log n+\frac 34\log\log\log n+\frac 12\log\log\log\log n+\cdots+(\frac 12+\delta)\log^{(k)}n)\qquad\text{for only finitely many n}$$ In particular, this implies that: $$\limsup_{n\to \infty} \frac{\frac{S_n}{\sqrt{2n\log\log n}}-1}{\frac{\log\log\log n}{\log\log n}}=\frac 34$$