I am trying to find a generalization of the convolution formula between distribution functions. In the case of 2 functions, I know that:
$$F_0*F_1\,(x) = \int_{-\infty}^{+\infty}F_0(x-x_1) d\,F_1(x_1) $$
Where $d\,F_1(x_1)$ stresses the Lebesgue-Stieltjes integral notation. I would like generalize to 3 functions. I can do:
$$F_0*F_1*F_2\,(x) = \int_{-\infty}^{+\infty}F_0(x-x_1) d\,F_1*F_2\,(x_1) $$
Where, according to the firts equation: $F_1*F_2 \,(x_1)= \int_{-\infty}^{+\infty}F_0(x_1-x_2) dF_2(x_2)$. But I don't know how to find:
$$d\, F_1*F_2\,(x_1) = ? $$
to after replace on the second equation and then find my forumula.
How can I do this?
Update:
I tryed something like this. Let $f$ be the density of some distrib. function $F$, then $ dF(x) = f(x)dx$. So, we can do $$d F_1*F_2(x_1) = f_1 * f_2 (x_1)dx_1$$ Where $f_1 * f_2 (x_1) = \int_{-\infty}^{+\infty}f_1(x_1 - x_2)f_2(x_2) dx_2$. So, \begin{align} F_0*F_1*F_2\,(x) &= \int_{-\infty}^{+\infty}F_0(x-x_1) d\,F_1*F_2\,(x_1)\\ &=\int_{-\infty}^{+\infty}F_0(x-x_1) f_1 * f_2 (x_1)dx_1\\ &=\int_{-\infty}^{+\infty}F_0(x-x_1) \left[\int_{-\infty}^{+\infty}f_1(x_1 - x_2)f_2(x_2) dx_2\right] dx_1\\ &=\int_{-\infty}^{+\infty}\left[\int_{-\infty}^{+\infty}F_0(x-x_1) f_1(x_1 - x_2)f_2(x_2) dx_2\right]dx_1\\ &=\int_{-\infty}^{+\infty}\left[\int_{-\infty}^{+\infty}F_0(x-x_1) f_1(x_1 - x_2)f_2(x_2) dx_1\right]dx_2\\ &=\int_{-\infty}^{+\infty}\left[\int_{-\infty}^{+\infty}F_0(x-x_1) f_1(x_1 - x_2) dx_1\right]f_2(x_2)dx_2\\ &=\int_{-\infty}^{+\infty}\left[\int_{-\infty}^{+\infty}F_0(x-x_1) f_1(x_1 - x_2) dx_1\right]d F_2 (x_2) \end{align} From here on, I don't know how to get a closed equation.