Consider the probability distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where
$\lambda\equiv (\lambda_1,...,\lambda_J)$
$\mu\equiv (\mu_1,...,\mu_J)$
$\mu_1<...<\mu_J$, $\mu_j\in \mathbb{R}$ $\forall j$
$\lambda_j\in [0,1]^J$ and $\sum_{j=1}^J \lambda_j=1$
Given $\Delta(\cdot ; \lambda, \mu)$, let $\Delta^{-}(\cdot; \lambda, \mu)$ denote $\Delta(\cdot ; \lambda, \mu)$ reflected over the origin, i.e., $$ \Delta^{-}(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\leq -\mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$
Question: Consider two random variables $Y,Y'$, stochastically independent, with $Y\sim \Delta(\cdot; \lambda, \mu)$ and $Y'\sim \Delta(\cdot; \lambda', \mu')$ (where "$\sim$" denotes distributed as). Is it correct to say that $$Y-Y'\sim \Delta(\cdot; \lambda, \mu)* \Delta^{-}(\cdot; \lambda', \mu')$$ where $*$ denotes the convolution operator?