Unfortunately, I am not very familiar with the field of probability distributions. However, I am trying to compute the probability density function of a random variable $Z=AX+\delta AX$, where $A \in \mathbb{R}^{n \times n}$, $\delta \sim p_\Delta$ and $x \sim p_X$.
I figured out that by the product rule of two random variables, say $Y:=\delta AX$ it holds $$ p_Y(y) = \int_{-\infty}^{\infty} p_X(x) p_\Delta(\frac{y}{x}) \frac{1}{\vert x \vert}\ dx. $$ Can we now apply the summation rule and write the sum in $Z$ as $$p_Z(z) = \int_{-\infty}^{\infty} p_X(s) p_Y(z-s) \ ds = \int_{-\infty}^{\infty} p_X(s) \int_{-\infty}^{\infty} p_X(x) p_\Delta(\frac{z-s}{x})\frac{1}{\vert x \vert} \ dxds$$ to obtain the distribution of $Z$?