Recovery of periodicities hidden in heavy‐tailed noise

Abstract

AbstractWe address a parametric joint detection‐estimation problem for discrete signals of the form , , with an additive noise represented by independent centered complex random variables . The distributions of 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 , their number N, and complex coefficients . For example, one of considered classes of noise is the following: are independent identically distributed random variables with and . 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.