let define a 2D random walk by
$$ \sum_i A_i X_i $$
where $A=[\cos(\theta)\ \sin(\theta)]^T$, $\theta$ is a random variable in the range $[0,2\pi]$ and $X$ is a scalar random variable between $[-1,1]$. $\theta$ and $X$ have uniform distribution and they are independent.
The random walk can be split to two random walks in $x$ and $y$. Then, we end up to two variances $\sigma_x$ and $\sigma_y$. How can we compute the variance of the 2D walk based on $\sigma_x$ and $\sigma_y$? Maybe $\sqrt{\sigma_x ^2 + \sigma_y ^2}$?
If we calculate the variance of the vector $AX$, we end up to a $2 \times 2$ covariance matrix. I assume this should end to an identical result. But how?