Recovery of periodicities hidden in heavy-tailed noise

Abstract

We address a parametric joint detection-estimation problem for discrete signals of the form $x(t) = \sum_{n=1}^{N} \alpha_n e^{-i \lambda_n t } + \epsilon_t$, $t \in \mathbb{N}$, with an additive noise represented by independent centered complex random variables $\epsilon_t$. The distributions of $\epsilon_t$ 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 $\lambda_n$, their number $N$, and complex amplitudes $\alpha_n$. For example, one of considered classes of noise is the following: $\epsilon_t$ are independent identically distributed random variables with $\mathbb{E} (\epsilon_t) = 0$ and $\mathbb{E} (|\epsilon_t| \ln |\epsilon_t|) < \infty$. 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.