Exercise:
Find a condition on the $a_{n}$ that guarantees the series $$\sum_{i=0}^{^\infty} a_{n}X_{n}$$ is almost surely convergent, where $X_{n} $ i.i.d. with density function: $$ g(x) = \begin{cases} C|x|^{-3/2} & \text{for |x| $\gt$ 1} \\[2ex] 0 & \text{for $|x| \le 1 $} \end{cases} $$
I try using to Kolmogorov's three-series theorem to prove them. The theorem states that the series $\sum_{n = 1}^\infty a_n X_n$ converges almost surely, iff all of three following conditions are satisfied:
- $\sum_{n=1}^{\infty}\mathbb{P}(|a_{n}X_n| \ge M)$ converges.
- Let $Y_n = X_n\mathbf{1}\{|a_{n}X_n|\le M\}$, then $\sum_{n=1}^{\infty}\mathbb{E}[Y_n]$, the series of expected value's of $Y_n$, converges.
- $\sum_{n=1}^{\infty}\mathrm{var}(Y_n)$ converges.
Edit: Proof: I will try to show the above three hold.
Let M = 1. For infinietly many $|a_{n}|>1$, this series is not converges. Supoose for specific $n_0$: $|a_{n}| < 1$. We Have $ \mathbb{P}(\sum_{n=n_{0}}^{\infty}|a_{n}X_n| \ge 1) = \sum_{n=n_{0}}^{\infty}\mathbb{P}(|X_{n}| \ge \frac{1}{|a_{n}|}) = \sum_{n=n_{0}}^{\infty}\int_{\frac{1}{|a_{n}|}}^{\infty} C|x|^{-3/2} dx = \sum_{n=n_{0}}^{\infty}\frac{C_{1}}{\sqrt{\frac{1}{|an|}}} = \sum_{n=n_{0}}^{\infty} C_{1}\sqrt{|a_{n}|} $
- If $\sum_{n=n_{0}}^{\infty} = \infty $, $ X_{n}, X_{n+1}$ events are independent, then Borel–Cantelli lemma with probability 1 $\{|a_{n}X_{n}| \le 1\}$ occur for infinitely many values of n. So series is diverges.
- If $\sum_{n=n_{0}}^{\infty} < \infty $. We have Borel–Cantelli lemma and $\{|a_{n}X_{n}| \le 1\}$ occur for finite many values of n. We have $\sum_{n=n_{0}}^{\infty}a_{n}X_n\mathbf{1}\{|a_{n}X_n|\le 1\} = 0$ and $\sum_{n=n_{0}}^{\infty}a_{n}^{2}EX_n^{2}\mathbf{1}\{|a_{n}X_n|\le 1\} = C\sum_{n=n_{0}}^{\infty}a_{n}^{2}\frac{1}{|a_{n}|^{3/2}} = C\sum_{n=n_{0}}^{\infty}\sqrt{|a_{n}|} < \infty $. So series is convergent.