Problem:
Let $\{B_n\}$ be a sequence of independent, identically distributed Bernoulli random variables of parameter $p=\frac12$. Let $\{a_n\}$ be a non-negative, non-random sequence such that $\sup\limits_{n\geq0}\frac1n\sum\limits_{i=1}^na_i^2<\infty$ and $\frac1n\sum\limits_{i=1}^{n}a_i$ converges. Prove that $\frac1n\sum\limits_{i=1}^na_i(B_i-\frac12)$ converges to zero almost surely.
This is an exercise in the section of "strong law of large numbers". But I only managed to prove the case when $\{a_n\}$ is bounded.
Any help is appreciated!!
There are loads of ways to go about this; I think the purpose of this problem is not to apply a strong law of large numbers theorem, but to essentially adapt the proof to this setting. There are many many ways to do this, but I'm gonna sketch out a sort of easy, cheap way.
Strategy: We're gonna let $S_n = \sum_{j = 1}^n a_j (B_j - 1/2)$ and show that $\mathbb{E}[S_n^4] \leq C n^2$ for some constant $C$. From here, we'll get that $\mathbb{E} [(S_n / n)^4]$ is summable, which will imply that $S_n/n \to 0$ almost surely.
Use the assumptions on $a_n$ to show that there exists a $C$ so that $a_j \leq C \sqrt{j}$ for all $j$.
By expanding directly, show that $\mathbb{E}(S_n^4) \leq c_1 \sum_{j = 1}^n a_j^4 + c_2 (\sum_{j = 1}^n a_j^2)^2$ for some constants $c_1,c_2$.
Use the assumption that $\frac{1}{n} \sum_{ j =1}^n a_j^2$ is bounded to complete the upper bound above.