Recovery of periodicities hidden in heavy-tailed noise

Abstract

We address a parametric joint detection-estimation problem for discrete signals of the form $x(t) = \sum_{n=1}^{N} α_n e^{-i λ_n t } + ε_t$, $t \in \mathbb{N}$, 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., the frequencies $λ_n$, their number $N$, and complex amplitudes $α_n$. For example, one of considered classes of noise is the following: $ε_t$ are independent identically distributed random variables with $\mathbb{E} (ε_t) = 0$ and $\mathbb{E} (|ε_t| \ln |ε_t|) < \infty$. 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. We discuss also decaying signals and the case of infinite number of frequencies.