Functional estimation of a density under a new weak dependence condition (Q2739868)

The authors analyze properties of usual kernel density estimators in the case where the data \(X_1\),\dots,\(X_n\),\dots is a time series satisfying a new weak dependence condition:NEWLINENEWLINENEWLINEfor any bounded \(f:R^n\to R\),\quad \(g: R^m\to R\),\quad \(i_1\leq\dots\leq i_n<i_n+r\leq j_1\leq\dots\leq j_m\), NEWLINE\[NEWLINE|\text{Cov}(f(X_{i_1},\dots,X_{i_n}),g(X_{j_1},\dots,X_{j_m}))|\leq C \text{Lip}(f)\text{Lip}(g)\vartheta_rNEWLINE\]NEWLINE or NEWLINE\[NEWLINE|{\text Cov}(f(X_{i_1},\dots,X_{i_n}),g(X_{j_1},\dots,X_{j_m}))|\leq C\min\{\text{Lip}(f),\text{Lip}(g)\}\vartheta_r,NEWLINE\]NEWLINE where \(\text{Lip}\) is the Lipschitz modulus and \(\vartheta_r\) is some fixed number sequence \(\vartheta_r\to 0\) as \(r\to\infty\). It is demonstrated that such inequalities can be derived with various \(\vartheta_r\) for Bernoulli shift sequences, e.g., for Volterra processes and ARMA and bilinear processes. The authors derive bias and MISE asymptotics, asymptotic normality results and a.s. convergence properties for kernel estimates under these mixing conditions. E.g., if \(\vartheta_r=O(r^{-12-\nu})\) then asymptotic normality holds true.