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=1NI±ne−iI»nt+Iµt, t∈N, with an additive noise represented by independent centered complex random variables Iµt. The distributions of Iµt 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 I»n, their number N, and complex coefficients I±n. For example, one of considered classes of noise is the following: Iµt are independent identically distributed random variables with E(Iµt)=0 and E(|Iµt|ln|Iµt|)