Statistical inference across time scales

Abstract

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=Δ_T relative to T in the limit T→∞. We quantify the smooth statistical transition from a microscopic Poissonian regime (when Δ_T→0) to a macroscopic Gaussian regime (when Δ_T→∞). 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 Δ_T→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.