Consider two independent Poisson processes with rates $\lambda_1$ and $\lambda_2$ that describe two types of events 1 and 2, respectively.
Let $T_{1,}$ indicate the instant of the $i$-th event of type 1 and $T_{2,j}$ the instant of the $j$-th events of type 2 over the timeline. Is it correct to claim that $min_{i,j}\{T_{2,i}-T_{1,j}|T_{2,i}-T_{1,j}>0\}$ (i.e., consecutive events), are exponentially distribute with rate $\lambda_1$, and that $min_{i,j}\{T_{1,i}-T_{2,j}|T_{1,i}-T_{2,j}>0\}$ for all $i$ and $j$ are exponentially distributed with rate $\lambda_2$?
It probably relates to the memoryless property of the exponential distribution if it is correct. But I am not sure. I know it is invalid if the events are not consecutive (then it is Erlang).
Can anyone help to analyze it, please? Or even share thoughts.