  We investigate statistical inference across time scales. We take as toy model
the estimation of the intensity of a discretely observed compound Poisson
process with symmetric Bernoulli jumps. We have data at different time scales:
microscopic, intermediate and macroscopic. We quantify the smooth statistical
transition from a microscopic Poissonian regime to a macroscopic Gaussian
regime. The classical quadratic variation estimator is efficient in both
microscopic and macroscopic scales but surprisingly shows a substantial loss of
information in the intermediate scale that can be explicitly related to the
sampling rate. We discuss the implications of these findings beyond this
idealised framework.
