Let $Q_{1}\left(t\right)=\sum_{i_{1}=1}^{N_{1}\left(t\right)}Y_{i_{1}}^{\left(1\right)}, Q_{2}\left(t\right)=\sum_{i_{2}=1}^{N_{2}\left(t\right)}Y_{i_{2}}^{\left(2\right)},\ldots,Q_{n}\left(t\right)=\sum_{i_{n}=1}^{N_{n}\left(t\right)}Y_{i_{n}}^{\left(n\right)}$ be independent compound Poisson-processes. Here $Y_{i_{j}}^{\left(j\right)}$ are different (or at least they can be different) distributed independent random variables, and they are independent from the $N_{j}$ Poisson processes as well. $N_{j}$ processes are also independent from each other. (Basically, everything is independent from each other.) Let me denote the sum of these compound Poisson processes as
$$Q\left(t\right)=Q_{1}\left(t\right)+Q_{2}\left(t\right)+\ldots+Q_{n}\left(t\right)=\sum_{i=1}^{N\left(t\right)}Y_{i},$$
where I know the sum is also a compound Poisson-process. I know in this general case, to determine the distribution of $Q\left(t\right)$ it is a standard method to use charachteristic/moment generating functions and after it to use some kind of inverse transformation, but do we know anything about the distribution of $Y_{i}$?
My conjecture is the following:
$$Y_{i}\overset{d}{=}\chi_{1}\cdot Y_{i_{1}}^{\left(1\right)}+\chi_{2}\cdot Y_{i_{2}}^{\left(2\right)}+\ldots+\chi_{n}\cdot Y_{i_{n}}^{\left(n\right)},$$
where $\chi_{j}$ are indicators. $\chi_{j}=1$ if the examined jump of $Q\left(t\right)$ is $Y_{i_{j}}^{\left(j\right)}$ and $0$ otherwise. I think if $N_{1},N_{2},\ldots,N_{n}$ have intensity parameters $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ respectively, then the $p_{j}$ parameters of $\chi_{j}$s are
$$p_{1}=\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}+\ldots+\lambda_{n}},p_{2}=\frac{\lambda_{2}}{\lambda_{1}+\lambda_{2}+\ldots+\lambda_{n}},\ldots,p_{n}=\frac{\lambda_{n}}{\lambda_{1}+\lambda_{2}+\ldots+\lambda_{n}}.$$
I have this conjecture, because if $\mu_{j}>\mu_{k}$ $j\neq k$, then $Y_{i_{j}}^{\left(j\right)}$ will occure more often, then $Y_{i_{k}}^{\left(k\right)}$, and I think the $\mu$ parameters are additive. If this conjecture was true, then from this point it would be easier to determine the distribution of $Y_{i}$.
Do you think is it true? Is there any case, when we have a closed form for the charachteristic function/PDF/CDF of $Y_{i}$?