Convergence of $S_n = \sum_{k=1}^{n} kX_k$ with $X_k$ to be mutually independent R.V.

Question

I am currently doing an exercise saying that :Let $\{X_k\}$ be mutually independent random variables, let $\{ v_k \}$ be non-random reals such that $E(X_k)= 0 $ and $Var(X_k) \leq v_k < \infty$ for each $k = 1,2,...$ and set $S_n/n = \sum_{k=1}^{n} kX_k$ Show that : 1) if $\sum_{k=1}^{\infty}v_k <\infty$ then $S_n/n$ converges to $0$ almost surely 2) if $\sum_{k=1}^{\infty}v_k = \infty$ and $\limsup X_n =1 $ almost surely, then $S_n/n$ converges to $\infty$ almost surely 3) if exist real $v$, $C$ and $\alpha < \frac{1}{2}$ such that $Var(X_k)=v$ and $|X_k| \leq Ck^{\alpha}$ for any k , then $S_n/n^{\frac{3}{2}}$ converges in distribution to some limiting distribution and find this distribution.

Can someone give me the brief idea on how to think about these questions.

Answer 1

For statement 1): Denote $Y_n=nX_n$, then by assumption, $(Y_n)$ is mutually independent and $$\sum_{n=1}^\infty \frac{Var Y_n}{n^2} <\infty.$$ It follows from Kolmogorov's strong law of large numbers that $$\frac{1}{n}\sum_{k=1}^n Y_k \to 0\;\mbox{ a.s. as } n\to \infty,$$ and this is the desired limit.