I'm considering approximating a Poisson mixture distribution with a simple Poisson distribution by using the mean, $\mu_\pi$, of the mixing distribution, $\pi$, as the rate parameter, $\lambda$, for the simple Poisson.
My question concerns the timescale of fluctuations in the rate parameter, $\lambda$, of the underlying process. If $\lambda$ fluctuates many times during the entire process (thus providing a good sample of the mixing distribution, $\pi$), but relatively few times between successive events (e.g., only one fluctuation for many events), can the approximation still be valid? Or are frequent fluctuations between events a necessary condition for the validity of this 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 the frequency of fluctuations.
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?). Could someone clarify the precise relationship between the timescale of $\lambda$ fluctuations, the time between events, and the validity of the simple Poisson approximation?
