L2(Rd)$\mathbb {L}_{2}(\mathbb {R}^d)$-Almost sure convergence for multivariate probability density estimate from dependent observations

Mohammed Badaoui and Noureddine Rhomari

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 3, 1306-1316

Abstract: We study the almost sure convergence of integrated square error of the wavelet density estimators for multivariate absolutely regular observations. We state that these estimates reach, up to a logarithm, the optimal rate of L2(Rd)$\mathbb {L}_{2}(\mathbb {R}^{d})$-almost sure convergence for densities in the Sobolev space H2s(Rd)$\mathbf {H}^{s}_2(\mathbb {R}^{d})$ with s > 0. The support of f may be the whole space Rd$\mathbb {R}^{d}$. Precisely, if fn is a such estimate of f, we prove that ∥fn-f∥L2(Rd)=O(nlogn)-sd+2s$\Vert f_n-f\Vert _{\mathbb {L}_{2}(\mathbb {R}^d)}=\mathcal {O}(\frac{n}{\log n})^{-\frac{s}{d+2s}}$, a.s. Moreover, we give an estimate of the constant in this upper bound. Proofs are based on Hilbertian approach and Bernstein type inequalities for dependent Hilbertian random vectors.

Date: 2017
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DOI: 10.1080/03610926.2015.1019139

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