Recovery of periodicities hidden in heavy†tailed noise

We address a parametric joint detection†estimation problem for discrete signals of the form x(t)=∑n=1Nαne−iλnt+εt, t∈N, with an additive noise represented by independent centered complex random variables εt. The distributions of εt 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 λn, their number N, and complex coefficients αn. For example, one of considered classes of noise is the following: εt are independent identically distributed random variables with E(εt)=0 and E(|εt|ln|εt|)