There are two vectors $\mathbf{a}$ and $\mathbf{b}$ of length $M$ where their entries follow the normal distribution, i.e., $a_i \sim \mathcal{N}(0, \alpha)$ and $b_i \sim \mathcal{N}(0, \beta)$. With the assumption $\alpha \neq \beta$, we have:
$$\mathbf{a}^T \mathbf{b} \rightarrow 0 \mbox{ when } M \rightarrow \infty$$
One textbook says that the convergence rate of $\mathbf{a}^T \mathbf{b} \rightarrow 0 $ is $\sqrt{M}$ but I do not know why they can come up with this rate. Do you have any idea?