Consider a compound Poisson process $$\sum_{i=1}^{N_t}X_i$$ where $N_t$ is a poisson process with rate $\lambda_1$ and $X_i$ i.i.d. with cumulative distribution function $F(.)$ so that mean and variance are both finite.
Now I need to find another r.v. $Z_j$ such that $$\sum_{j=1}^{M_t} Z_j= \sum_{i=1}^{N_t} X_i$$ where $M_t$ is another poisson process defined on the same $t$ with rate $\lambda_2$.
$Z_j$ should be i.i.d. with mean and variance finite. Does $Z_j$ exist? If so, how much information can we figure out about its cdf ?
Thx here for any helping hand.