Reading a book a find this results: - Assume we have a logistic random variable $X$ (with parameters $(\lambda, \theta)$) with pdf \begin{eqnarray*} f (x) &=& \displaystyle \frac {\displaystyle \lambda \exp \left\{ -\lambda \left( x - \theta \right) \right\}} {\displaystyle \left[ 1 + \exp \left\{ -\lambda \left( x - \theta \right) \right\} \right]^2}\,. \end{eqnarray*} The last pdf can be re-expressed as a mixture of Laplace distribution \begin{eqnarray} f(x) &=& \displaystyle \sum_{k = 0}^\infty \frac {2}{k + 1} {-2 \choose k} \frac {\lambda (k + 1)}{2} \exp \left\{ -\lambda (k + 1) \mid x - \theta \mid \right\}. \label{ffx} \end{eqnarray} Now, suppose $X$ and $Y$ are independent logistic random variables with parameters $(\lambda, \theta)$ and $(\mu, \phi)$, respectively. The cdf of $Z = \alpha X + \beta Y$ can be expressed as: \begin{eqnarray} \Pr \left( \alpha X + \beta Y \leq z \right) &=& \displaystyle \sum_{k = 0}^\infty \sum_{l = 0}^\infty \frac {\displaystyle 4}{\displaystyle (k + 1) (l + 1)} {-2 \choose k} {-2 \choose l} \Pr \left( \alpha X_k + \beta Y_l \leq z \right), \end{eqnarray} where $X_k$ and $Y_l$ are independent Laplace random variable with parameters $(\lambda (k + 1), \theta)$ and $(\mu (l + 1), \phi)$, respectively.
Please help me to understand how the last, double infinite sum, representation is obtained?
This is just swapping integration with summation. For any region $D$ in the plane we have $$ P((X,Y)\in D)= \iint_D f_X(x)f_Y(y)dx\,dy. $$ Plug in the representation of $f_X$ as a mixture of $f_{X_k}$: $$ f_X(x)=\sum_{k=0}^\infty\frac2{k+1}{-2\choose k}f_{X_k}(x) $$ (note the $\frac{\lambda(k+1)}2$ is missing because it's part of the Laplace density), and similarly: $$ f_Y(x)=\sum_{l=0}^\infty\frac2{l+1}{-2\choose l}f_{Y_l}(y) $$ and obtain $$ \begin{align} P((X,Y)\in D)&= \iint_D \sum_{k=0}^\infty\frac2{k+1}{-2\choose k}f_{X_k}(x)\sum_{l=0}^\infty\frac2{l+1}{-2\choose l}f_{Y_l}(y)dx\,dy\\ &=\sum_{k=0}^\infty\frac2{k+1}{-2\choose k}\sum_{l=0}^\infty\frac2{l+1}{-2\choose l}\iint_D f_{X_k}(x) f_{Y_l}(y)dx\,dy\\ &=\sum_{k=0}^\infty\sum_{l=0}^\infty\frac2{k+1}{-2\choose k}\frac2{l+1}{-2\choose l}P((X_k,Y_l)\in D). \end{align} $$