# Compound and Thinning Poisson Process

I made a question here Poisson Distribution with Changing Lambda? where I was curious about Poisson Arrival Process where the lambda parameter varies according to an AR process or a discrete markov process.

I was wondering : Could any of these plots be considered as a Thinning Process or a Compounded Process?

1) Thinning Process:

Let $$N$$ be the number of events in the original Poisson process with rate parameter $$\lambda$$. The Poisson thinning process $$N(t)$$ is given by:

$$N(t) = \sum_{i=1}^{N} X_i$$

where:

• $$X_i$$ is a Bernoulli random variable that is 1 with probability $$g(t_i)$$ and 0 with probability $$1 - g(t_i)$$,
• $$t_i$$ is the time of the $$i$$-th event in the original Poisson process.
• $$g(t)$$ is a probability distribution function that varies with time $$t$$.

For each event in the original Poisson process, a decision is made whether to retain it or discard it based on the probability distribution $$g(t)$$. The resulting process, $$N(t)$$, is a thinned version of the original Poisson process.

2) Compound Process:

Let $$N(t)$$ be a Poisson process with rate $$\lambda$$, representing the number of events occurring in the time interval $$[0, t]$$.

For each event in the Poisson process, a random variable $$X_i$$ is the intensity of the $$i$$-th event. The random variables $$X_1, X_2, \ldots$$ are independent and identically distributed (i.i.d.).

The compound Poisson process $$S(t)$$, is defined as the sum of all events up to time $$t$$:

$$S(t) = \sum_{i=1}^{N(t)} X_i$$