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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.

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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 $t$:

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

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