Solved – Superposition of two dependent Poisson processes

\$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.)

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 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).

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