Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $\{X_n: n\ge 1\}$ be sequence of square integrable random variables, i.e., $X_n \in L^2(\Omega, \mathcal{F}, \mathbb{P})$ for each $n\ge 1$. Further assume that $\mathbb{E}[X_i X_j] = 0$ whenever $i\neq j$ and $\sup_n \mathbb{E}[X_n^2] < \infty$. For each $n \ge 1 $ set $S_n := \sum_{j=1}^n X_j$.
I am trying to show that:
(i) for each $\alpha > \frac{1}{2}$, $\frac{S_n}{n^{\alpha}}\rightarrow 0$ in probability.
(ii) for each $\alpha > 1$, $\frac{S_n}{n^{\alpha}}\rightarrow 0$ a.s.
(iii) $\{\frac{S_n}{n}: n \in\mathbb{N}\}$ is uniformly integrable.
My questions:
In part (i) and (ii) is the assumption "$\mathbb{E}(X_n) = 0$ for each $n$" missing? Or these results hold without this extra assumption?
Note that the said assumption, along with $\mathbb{E}[X_i X_j] = 0$ imply that $X_j$'s are uncorrelated and hence one can use
an argument similar to the proof of WLLN (Chebyshev) to show $\mathbb{P}\left(\frac{|S_n|}{n^\alpha}\ge \epsilon\right) \le \frac{n \sup_n \mathbb{E}[X_n^2]}{n^{2\alpha}\epsilon^2}, \alpha > 1/2$ for part (i), and
an argument similar to the proof of SLLN (Rajchman) to show $\mathbb{P}\left(\frac{|S_n|}{n^\alpha}\ge \epsilon \text{ infinitely often}\right) \le \mathbb{P}\left(\frac{|S_n|}{n}\ge \epsilon \text{ infinitely often}\right) = 0, \alpha >1$.
If the mentioned assumption is not indeed necessary, how can one go about solving this problem? Also any hint for part (iii) is appreciated.