# superposition of infinitely many poisson processes

I know that the superposition of two Poisson process with rates $$\lambda_1$$ and $$\lambda_2$$ is again a Poisson process with rate $$\lambda_1+\lambda_2$$. Thus this process has interarrival times distributed as an exponential with parameter $$\lambda_1+\lambda_2$$.

What I am actually doing is numerically sampling the exponential distribution: I collect a bunch of random times sampled from $$\lambda e^{-\lambda t}$$. The point is that I do it for different values of $$\lambda$$ and put all together in the same file. Then I plot the distribution. Let's assume the simple case in which I sample an equal number of times for each value of $$\lambda$$ and that I make $$\lambda$$ vary say from $$10^{-4}$$ to $$10^3$$.

To my intuition the resulting interevent time distribution should be $$$$\int_{\lambda_{min}}^{\lambda_{max}} P(\lambda)\lambda e^{-\lambda t} d\lambda.$$$$ Given the above hypothesis of uniformly distributed $$\lambda$$ values I just have $$P(\lambda) = const$$ and the integral is then $$$$\frac{1}{t}\left(\lambda_{min} e^{-\lambda_{min} t} - \lambda_{max} e^{-\lambda_{max} t}\right) + \frac{1}{t^2}\left(e^{-\lambda_{min} t} - e^{-\lambda_{max} t}\right).$$$$

Looking (and plotting) the above function, I expect two different regimes, one in which the interevent time distribution decays as $$t^{-1}$$ (large times) and one in which decays as $$t^{-2}$$ (small times). I don't find the second regime in the data.

Question: is it my approach that is wrong or the integral I am doing should actually give me the observed interevent time distribution? If it's correct, why the data don't show the small time regime?