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

We consider centered compound Poisson processes with finite variance, discretely observed over [0, T ] and let the sampling rate ∆ = ∆T → ∞ as T → ∞. 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 ∆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 ∆T grows faster than √ T . We also prove an asymptotic equivalence result, from which we identify, for instance, regimes where the increments cannot be distinguished from Gaussian variables. AMS subject classifications: 62B15, 62K99, 62M99.