Let $X_1$ and $X_2$ be independent Poisson random variates with parameters $\lambda_1$ and $\lambda_2$, respectively. One can show that $$ \mathbb{P}(X_1\ge X_2) \ge \mathbb{P}(X_2\ge X_1)\tag{1} $$ if and only if $$ \lambda_1\ge\lambda_2.\tag{2} $$ Since the parameters are real and real numbers are equipped with total order we can use the above equivalence to lift total ordering to (independent) Poisson random variates. So far so good.
Question: can we generalize this idea of "inheriting" total order from the reals to independent compound Poisson random variates?
I tried to work out the probabilities above for compound Poisson RVs and compare the two sides of (1), but couldn't see any scalar combination of the respective parameters forming that could replace/generalize (2). I wonder if this inheritance makes sense at all in this case.