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

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.