The problem is that my distribution gives the first incidence rate, but not the incidence rate. As soon as $S$ is incident on the boundary, we have to restart the process, hence we need to convolve $k$ Paretos.

Thanks a lot! Well, I am probably using Pareto Type I. I tried to model the rate of incidence on a boundary in a random walk, $S$. Assume boundary is at x=0 and you discretised the space as 0.5, 1.5, 2.5, 3.5... So, if $S$ is at $x=0.5$ at $t=t$, if it moves towards the boundary, it will be reflected and be back on $x=0.5$ at $t=t+1$. I didn't even know Pareto distribution could be used to model this, I didn't even know there is a distribution called Pareto :) I arrived at Pareto distribution using the Bertnard's Ballot Theorem.

Just to bother you one last time, I assume multivariate Pareto distribution or generalised Pareto distribution are not the thing I am looking for, right? There are limited documentation on them and they don't explicitly state that multivariate Pareto dist is addition of Pareto random variables and they are too tedious to understand :)

Can you tell me how this might work for three or more pareto distributions? Is it still a ${}_{2}F_{1}$, or does it change to some other generalised hypergeometric function? Also, does this distribution have a name, like Erlang is to Exponential?