## Abstract

We address a parametric joint detection-estimation problem for discrete signals of the form x(t) = Σ^{N}
_{n=1}α_{n}e^{-iλnt} + ϵ_{t}, t ∈ 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., frequencies λ_{n}, their number N, and complex coefficients α_{n}. For example, one of considered classes of noise is the following: ϵ_{t} are independent identically distributed random variables with E(ϵ_{t})=0 and E𝔼(|ϵ_{t}|ln|ϵ_{t}|) < ∞. 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.

Original language | English |
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Journal | Mathematische Nachrichten |

Volume | 291 |

Issue number | 1 |

Pages (from-to) | 86-102 |

Number of pages | 17 |

ISSN | 0025-584X |

DOIs | |

Publication status | Published - 01.01.2018 |