Cantelli's Law of Large Numbers

Question

In Probability, Shiryaev Chapter IV, Section 3 there's the following theorem:

Let $ X_1, X_2,... $ be indipendent random variables with finite fourth moments and let $ \mathbb{E}[|X_n - \mathbb{E}[X_n]|^4] \leq C $ for $ n \geq 1 $ and $ C $ some constant. Then, as $ n \rightarrow \infty $,

$ \frac{S_n - \mathbb{E}[S_n]}{n} \rightarrow 0 $ almost surely (where $ S_n = X_1 + ... + X_n) $.

The proof is the following:

WLOG, $ \mathbb{E}[X_n] = 0 \,\forall n $, so thesis is now $ \frac{S_n}{n} \rightarrow 0 $ almost surely. This follows if $ \sum \mathbb{P}(|\frac{S_n}{n}| \geq \varepsilon) < \infty, \,\forall \varepsilon > 0 $.

Until this bit, it's all good. Then it says:

"by Chebyshev's inequality, this will follow from $ \sum \mathbb{E}[|\frac{S_n}{n}|^4] \ < \infty $", and the proof goes on.

I can't see where that "4" comes from; I mean, Cheb's inequality says $ \mathbb{P}(|\frac{S_n}{n}| \geq \varepsilon) \leq \frac{Var(|\frac{S_n}{n}|)}{\varepsilon^2} = \frac{\mathbb{E}(|\frac{S_n}{n}|^2)}{\varepsilon^2} $ (assuming $ \mathbb{E}[X_n] = 0 $), so I sould show that $ \sum \mathbb{E}[|\frac{S_n}{n}|^2] \ < \infty $.

Somebody can tell me where I'm wrong? Thanks

Answer 1

$\sum \frac {E(S_n^{2})} {{n^{2}}}=\sum \frac {n Var X_1} {n^{2}}=\infty$ so you have to use the $4-$th power.

[$P(|Y| >c)\le \frac {E|Y|^{p}} {c^{p}}$ for any $p \in [1,\infty)$. Take $p=4$].