Interarrival Times in Nonhomogeneous Poisson Processes

Summary

The document contrasts homogeneous and nonhomogeneous Poisson counting processes through their event waiting times. For a constant rate, it notes that interarrival times are independent and exponentially distributed. It then asks what waiting-time assumptions produce a nonhomogeneous Poisson process, where the event intensity varies deterministically over time.

The text frames this as an unresolved question rather than presenting a derivation or answer. A useful distinction is that nonhomogeneous Poisson arrivals generally do not have identically distributed exponential gaps in ordinary time; their conditional waiting-time distribution depends on the integrated intensity. The document supplies no examples, applications to trading, or empirical evidence, so it serves mainly as a prompt about the relationship between event rates and waiting-time distributions.

Key ideas

• With a constant intensity, a homogeneous Poisson process has independent exponential interarrival times.
• A nonhomogeneous Poisson process allows deterministic intensity to vary with time.
• The document asks how waiting-time distributions relate to a nonhomogeneous counting process but does not provide the answer.
• For a varying intensity, waiting times in ordinary time are generally not identically distributed exponential variables.

Tags

# (Non-)Homogeneous Poisson process and their corresponding inter-arrival time distribution:


# (Non-)Homogeneous Poisson process and their corresponding inter-arrival time distribution:












When it is said that the number of claims follows a homogenous Poisson process (where the intensity is assumed to be a fix value), it means that we have the stationary and independent assumptions for the increments, and that the waiting times are exponentially distributed and independent.

In summary:

$N(t)\sim Poisson(\lambda)$, then the waiting time between two successive events $W_i \stackrel{i.i.d}{\sim} Exp(\lambda)$.

If the number of claims follows a non-homogeneous Poisson process (where the intensity is assumed to be a deterministic function of time), the waiting times are not exponentially distributed and independent.

In summary:

$N(t) \sim Poisson(\lambda(t))$, then we cannot say that $W_i$ are exponentially distributed. My question is that under which distributional assumption for the waiting time, the resulted counting process $N(t)$ would be a non-homogeneous Poisson process? From the above explanation, we know that if we want to reach a homogeneous Poisson process, we need to assume that the waiting time (or inter-arrival time distribution) follows exponential distribution, but what we can say in case we have a non-homogeneous Poisson process?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.