Recovery of periodicities hidden in heavy-tailed noise

We address a parametric joint detection-estimation problem for discrete signals of the form x(t) = Σ^N _n=1α_ne^-iλnt + ϵ_t, t ∈ N, with an additive noise represented by independent centered complex random variables ϵ_t. The distributions of ϵ_t are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy-tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., frequencies λ_n, their number N, and complex coefficients α_n. For example, one of considered classes of noise is the following: ϵ_t are independent identically distributed random variables with E(ϵ_t)=0 and E𝔼(|ϵ_t|ln|ϵ_t|) < ∞. The construction of estimators is based on detection of singularities of anti-derivatives for Z-transforms and on a two-level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series.