Sums of independent random variables (more than two)

Question

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$

What is the general formula for more than two RVs? For example, for three RVs.

Answer 1

Try something like $$\int_y \int_x f(z-x-y) \,g(y) \,h(x) \, dx\, dy$$ and this applies to independent random variables, not necessarily identically distributed.

Answer 2

Let $X,Y,Z$ be real-valued and independant random variables with density functions $f,g,h$. Then $X+Y+Z = (X+Y) + Z$.

Consider $X+Y$ as a random variable with density function $f*g$ and you now have a sum of two random variables. Therefore the general formula for the density function is just $(f*g)*h$. It can be shown that this is indeed associative, which mean that $(f*g)*h = f*(g*h)$.