I need your expertise in solving the following problem:
Let $n \in \mathbb{N}$ such that $n > 1$, $\left\lbrace X_i \right\rbrace_{i=1}^n$ be a set of sub-Gaussian variables with mean $0$ and unit variance and $\left\lbrace Y_i \right\rbrace_{i=1}^n$ be a set of random variables where for each $i \in [n]$, $Y_i \in \left\lbrace-1,1 \right\rbrace$. ($Y_i$ is dependent on $X_i$ for each $i \in [n]$)
We want to have bounds on the expectation of the dot product of $X$'s and $Y'$s i.e. $$ E\big[ \sum\limits_{i=1}^n x_iy_i \big]$$
How can we establish bounds on this?
Please advise.