Nonparametric estimation of periodic signal disturbed by α-stable noises

Abstract

The paper is concerned with the nonparametric estimation problem for periodic signal disturbed by α-stable noise based on a periodic kernel. The consistency and the inconsistency of the nonparametric estimator separately in terms of the range of α, i.e. α ∈ (1,2) and α ∈ (0,1] are discussed by using Markov inequality and the moment inequalities for stable stochastic integrals. Then, we state the strong consistency of the nonparametric estimator by using the triangular kernel and the Borel-Cantelli lemma when α ∈ (1,2). Besides, the asymptotic distributions of the nonparametric estimator are studied by applying the inner clock property for the α-stable stochastic integral. Taylor's formula and Slutsky's theorem. Finally, a numerical example is introduced to illustrate our theory