Question

Consider a nonhomogeneous Poisson process whose intensity function \(\lambda(t)\) is bounded and continuous. Show that such a process is equivalent to a process of counted events from a (homogeneous) Poisson process having rate \(\lambda\), where an event at time \(t\) is counted (independent of the past) with probability \(\lambda(t) / \lambda ;\) and where \(\lambda\) is chosen so that \(\lambda(s)<\lambda\) for all \(s\).

Short answer

We can show that a nonhomogeneous Poisson process with intensity function λ(t) is equivalent to counting events from a homogeneous Poisson process with rate λ by satisfying the following properties: 1. The number of events in non-overlapping time intervals are independent in both processes. 2. An event at time t is counted with probability λ(t)/λ, independent of past events, ensuring consistency with the rate function of the nonhomogeneous Poisson process. This equivalence is guaranteed by choosing λ such that λ(s) < λ for all s, where λ is the upper bound for any intensity rate in the nonhomogeneous Poisson process.

Step-by-step solution

Define a nonhomogeneous Poisson process

A nonhomogeneous Poisson process, also called an inhomogeneous Poisson process, is a counting process that generalizes the Poisson process to allow for varying intensity rates. The intensity function (or rate function) of a nonhomogeneous Poisson process is denoted by λ(t), where λ(t) is the instantaneous event rate at time t.

Define the homogeneous Poisson process

A homogeneous Poisson process is a special case of a nonhomogeneous Poisson process, which has a constant intensity function λ, where λ is a positive constant. The defining property of a homogeneous Poisson process is that the number of events in non-overlapping time intervals are independent and have a Poisson distribution with mean λ times the length of the interval.

Define probability of counting an event at time t

We are given that an event at time t is counted with probability λ(t) / λ, independent of past events. This implies that we have a thinning process that selects events from the original homogeneous Poisson process with rate λ by applying a probability of acceptance for each event at time t, given by the ratio of the nonhomogeneous intensity function to the homogeneous intensity.

Choose λ such that λ(s) < λ for all s

In order to ensure the equivalence between the nonhomogeneous Poisson process and the homogeneous Poisson process, we need to choose the λ of the homogeneous Poisson process such that for any time s, the intensity of the nonhomogeneous Poisson process at that time is less than the homogeneous intensity, i.e. λ(s) < λ. This essentially guarantees that the homogeneous rate is an upper bound for any intensity rate in the nonhomogeneous Poisson process.

Show equivalence between nonhomogeneous and homogeneous Poisson processes

Since we have a thinning process that selects events from a homogeneous Poisson process with rate λ and counts an event at time t with probability λ(t) / λ, independent of past events, we can show the equivalence of these processes through their respective properties: 1. The number of events in non-overlapping time intervals are independent in both processes. 2. The probability of an event being counted at any given time t is given by the ratio of nonhomogeneous intensity to homogeneous intensity, λ(t)/λ. This ensures that the probability of an event being counted is consistent with the rate function of the nonhomogeneous Poisson process. By satisfying these properties, we have shown that a nonhomogeneous Poisson process with intensity function λ(t) is equivalent to a process of counting events from a homogeneous Poisson process having rate λ, where an event at time t is counted with probability λ(t)/λ and where λ is chosen such that λ(s) < λ for all s.