Independence of Thinned Processes

Theorem

Let $\{N(t),t\ge 0\}$ be a Poisson process of rate $\lambda$. Each event is independently classified as type I with probability $p$ or type II with probability $1-p$.

Let $N_1(t)$ and $N_2(t)$ count type I and type II events respectively. Then:

1. $\{N_1(t),t\ge 0\}$ is a Poisson process with rate $\lambda p$
2. $\{N_2(t),t\ge 0\}$ is a Poisson process with rate $\lambda (1-p)$
3. $N_1$ and $N_2$ are independent

Interpretation

Splitting a Poisson process randomly produces two independent Poisson processes. The independence is non-obvious — you might expect that more type I events means fewer type II events, but the randomness of the total count balances this out.

Proof Sketch

Verify $\{N_1(t)\}$ satisfies the axiomatic definition of a Poisson process:

1. $N_1(0)=0$ (inherited from $N(0)=0$)
2. Independent and stationary increments inherited from $N(t)$
3. $P(N_1(h)=1)=p(\lambda h+o(h))+o(h)=\lambda ph+o(h)$
4. $P(N_1(h)\ge 2)\le P(N(h)\ge 2)=o(h)$

Independence follows because the classification of each event is independent of everything else, so knowledge of type II event times gives no information about type I events.

Example

Immigrants arrive at rate 10/week. Each is of English descent with probability $1/12$. The number of English immigrants in February (4 weeks) is $\text{Po}(4\cdot 10\cdot 1/12)=\text{Po}(10/3)$.

$P(\text{no English immigrants})=e^{-10/3}$

Related

• Poisson Process
• Thinning