When is it no longer possible to estimate a compound Poisson process?

Abstract

We consider centered compound Poisson processes with finite variance, discretely observed over $[0,T]$ and let the sampling rate $\Delta=\Delta_{T}\rightarrow\infty$ as $T\rightarrow\infty$. From the central limit theorem, the law of each increment converges to a Gaussian variable. Then, it should not be possible to estimate more than one parameter at the limit. First, from the study of a parametric example we identify two regimes for $\Delta_{T}$ and we observe how the Fisher information degenerates. Then, we generalize these results to the class of compound Poisson processes. We establish a lower bound showing that consistent estimation is impossible when $\Delta_{T}$ grows faster than $\sqrt{T}$. We also prove an asymptotic equivalence result, from which we identify, for instance, regimes where the increments cannot be distinguished from Gaussian variables.