Strong law of large numbers implications

Question

Let $S_n$ be the sum of i.i.d. components $X_i$ with $P(X_i= 0) = P(X_i = 1) = 1/2$. Then, according to the strong law of large numbers we have that $$ \frac{S_n}{n} \to 1/2 \quad a.s.$$ Can I follow that: Let $\epsilon > 0$. There exists a $n_0$ such that for all $n \geq n_0$: $$ \frac{n}{2} - \epsilon < S_n < \frac{n}{2} + \epsilon \quad a.s. \ $$ Or am I interpreting the SLLN incorrectly? Or, can I only infer a statement like $$ \frac{n}{4} < S_n < \frac{3n}{4} \quad a.s. \quad ?$$

Answer 1

The convergence delivered by the SLLN is point-wise, not uniform. The central limit theorem tells us that deviations of $S_n$ from $n/2$ are typically of size $\pm\sqrt n$, so your first formula asks for the impossible. The law of the iterated logarithm tells us that with probability $1$ we have

$$\limsup_{n\to\infty}\frac{S_n-n/2}{\sqrt{n\log\log n}}=1$$ and $$\liminf_{n\to\infty}\frac{S_n-n/2}{\sqrt{n\log\log n}}=-1.$$ This shows your first formula cannot hold, and gives a way to prove your second.