Solved – Superposition of two dependent Poisson processes

Question

$N_1(t)$ and $N_2(t)$ are two independent Poisson processes with intensities $λ_1$ and $λ_2$ respectively. $v_1$ and $v_2$ are two dependent positive random variables, and $p=P(v_1<c)=P(v_2<c)$, where $c$ is a constant.

When some random event of $N_1(t)$ happened and $v1<c$, it means an event of point process $N_3(t)$ happened. When some random event happened in $N_2(t)$ and $v_2<c$, it means an event of point process $N_4(t)$ happened.

Then $N_3(t)$ and $N_4(t)$ are two dependent Poisson processes with intensities $p*λ_1$ and $p*λ_2$ respectively. My questions are:

1. Is the superposition of $N_3(t)$ and $N_4(t)$ a Poisson process approximately?
2. Let $N_5(t)$ be a Poisson process with intensity $p(λ_1+λ_2)$, is there
   $P\{N_3(t)+N_4(t)\le m\} \le P\{N_5(t)\le m\}$? (Here $m$ is a integer.)

All your answers are helpful. I really appreciate them and thanks for gung's editing.

Specially, $v_1$ and $v_2$ be denoted as two dependent log-normal varialbles with correlation $\rho$. In other words, we can describe $v_1$ and $v_2$ as a multivariate log-normal variable. In this time, how to answer those two questions?

I agree that ($v_1$, $v_2$) can be replaced by the dependent Bernoulli pair($b_1$, $b_2$). And now $N_{3}(t)$ and $N_{4}(t)$ are two dependent poisson process. Is their superposition a poisson process approximately?

I really appreciate all your reply.

I have read sveral literature and consulted a professor. $N_{3}(t)$+$N_{4}(t)$ is a Cox process according to reply of the professor. And it converges to a poisson process.(Serfozo 1984, Chen & Xia 2011).

Answer 1

Too long for a comment.

But I do not think v1 and v2 can be replaced with a Bernoulli random variable. Specially, v1 and v2 be denoted as two dependent log-normal varialbles with correlation ρ. In other words, we can describe v1 and v2 as a multivariate log-normal variable. In this time, how to answer those two questions?

I think you're mistaken; they can be.

The only way that $v_1$ and $v_2$ affect anything is through the condition that they're less than $c$. Let $b_1$ and $b_2$ take the value 1 when their corresponding $v_i<c$. You now have that $b_1$ and $b_2$ are dependent Bernoullis. What information relevant to the question other than the information in the pair $(b_1,b_2)$ is there in $(v_1,v_2)$? If there is no additional information of relevance to the question, then $(b_1,b_2)$ clearly can replace $(v_1,v_2)$.