Edited: I'm considering using a simple Poisson distribution to model a process where the instantaneous rate of events, denoted by $\lambda$, is not constant. Instead, $\lambda$ depends on the current state of a complex underlying system with many interacting factors that change rapidly over time.
Due to the complexity of the system, I'm exploring the possibility of approximating the process with a simple Poisson distribution, where the rate parameter $\lambda$ is set to the time-averaged mean, $\mu_\pi$, of the fluctuating instantaneous rate. My primary concern is understanding how the timescale of fluctuations in $\lambda$ affects to this approximation.
While I don't have a precise model for the dynamics of the underlying system, I know that the factors influencing $\lambda$ change many times during the entire duration of the process, providing a good sample of the distribution of instantaneous rates. However, I'm unsure whether the number of these changes between successive events also plays a crucial role.
Specifically, I'm interested in understanding how the frequency/celerity of $\lambda$ fluctuations between successive events affects the accuracy of the simple Poisson approximation. If $\lambda$ fluctuates many times during the entire process (thus providing a good sample of the distribution of instantaneous rates, which I denote as $\pi$), but relatively few times between successive events, does this impact the quality of the poisson approximation?
According to Wikipedia (link to Mixed Poisson distribution), the variance of a mixed Poisson distribution is given by: $\operatorname{Var}(X) = \mu_\pi+\sigma_\pi^2$ where $\mu_\pi$ and $\sigma_\pi^{2}$ are the mean and variance of $\pi$, respectively. Based on this formula, one might expect the variance of the mixed process to approach this theoretical value as long as there are enough fluctuations of $\lambda$ to provide a good sample of $\pi$, regardless of whether there are many or few fluctuations between individual events. However, I performed some numerical simulations that seem to suggest that the frequency of fluctuations also plays a significant role in the variance (maybe some mistake in my code?). It seems that the approximation systematically overestimate the variance, are there other biases introduced?
