Long time ago, I was given the following exercise in a class.
Let $$X_1, X_2, \dots, X_n, \dots,$$ be a sequence of random variables, such that $\mathbb{E}X_iX_j =0$ for $i\not=j$ and $\mathbb{E}X_i^2 =1$ for all $i$. Let $$a_1,a_2,\dots, a_N,$$ be a sequence of real numbers. Prove that for $$Y_N:= \ \max_{i\le N} \Big|\sum_{k=1}^{i}a_k X_k\Big|,$$ we have $$\mathbb{E}Y_N^2 \ \le \ O(\log^2N)\cdot\sum_{k=1}^{N}a_k^2.$$ Unfortunately, I never learned how to solve this problem. It looks elementary, but I don't know how to attack it. Does anybody know how to do it? I will be grateful for any insight.
I don't really know either, but i will try to start.
Taking $N$ as you gave it, there exists an index $\overline{i}$ such that
$$\mathbb{E}Y_N^2=\mathbb{E}\bigl{\{}\sum_{k=1}^{\overline{i}}a_kX_k \bigr{\}}^2\le\mathbb{E}\bigl{\{}\sum_{k=1}^{\overline{i}}X_k \bigr{\}}^2\biggl{\{}\sum_{k=1}^{\overline{i}}a_k\biggr{\}}^2\le \mathbb{E}\bigl{\{}\sum_{k=1}^{\overline{i}}X_k \bigr{\}}^2\sum_{k=1}^{N}a_k^2$$ and now it would be convenient to show that $$\lim_{N\to +\infty}\frac{\mathbb{E}\bigl{\{}\sum_{k=1}^{\overline{i}}X_k \bigr{\}}^2}{log^2(N)}=0$$
Anyways, this is merely a way for you to be inspired, nothing else.