Statistical inference across time scales

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.