We consider a compound Poisson process with symmetric Bernoulli jumps, observed at times iΔ for i=0,1,… over [0,T], for different sizes of Δ=ΔT relative to T in the limit T→∞. We quantify the smooth statistical transition from a microscopic Poissonian regime (when ΔT→0) to a macroscopic Gaussian regime (when ΔT→∞). The classical quadratic variation estimator is efficient for estimating the intensity of the Poisson process in both microscopic and macroscopic scales but surprisingly, it shows a substantial loss of information in the intermediate scale ΔT→Δ∞∈(0,∞). This loss can be explicitly related to Δ∞. We provide an estimator that is efficient simultaneously in microscopic, intermediate and macroscopic regimes. We discuss the implications of these findings beyond this idealised framework.
"Statistical inference across time scales." Electron. J. Statist. 5 2004 - 2030, 2011. https://doi.org/10.1214/11-EJS660