Suppose that $x_1, \dots, x_n$ are i.i.d. with $x_i \sim N(0,I_k)$. Let $A_1, \dots, A_n$ be matrices with dimension $k \times k$ and $\|A_i\|_2 \leq 1$. Consider the following random vector
$$y = \sum_{i=1}^n A_i e_i $$
This vector is a multivariate Gaussian with mean zero and the covariance is given by
$$E[yy^T] = \sum_{i=1}^n A_i A_i^T$$
How can we compute a high probability upper bound for $\|y\|_2^2$?
My attempt:
Let $A = [A_1, \dots, A_n]$ and $E = [e_1^\top, \dots, e_n^\top]^\top$. We can write
$$ y = AE \Rightarrow \|y\|_2^2 \leq \|A\|_2^2 \|E\|_2^2$$
We have $\|A\|^2_2 \leq \sum_i \|A\|^2_i \leq n$ and $\|E\|_2^2 \sim \mathcal{X}(n)$ with a high probability upper bound given here. This approach gives an upper bound of $n^2$ but I think the correct bound should be around $n$ because the covariance of $y$ grows with $n$.