Wavelet-based estimators of partial derivatives of a multivariate density function for discrete stationary and ergodic processes

In this work, we propose a wavelet-based framework for estimating the derivatives of a density function in the setting of discrete, stationary, and ergodic processes. Our primary focus is the derivation of the integrated mean square error (IMSE) over compact subsets of ${\mathbb{R}}^d$, which provides a quantitative measure of estimation accuracy. In addition, the uniform convergence with rate and the normality are established. To establish the asymptotic behavior of the proposed estimators, we adopt a martingale approach that accommodates the ergodic nature of the underlying processes. Importantly, beyond ergodicity, our analysis does not require additional assumptions on the data. By demonstrating that the wavelet methodology remains robust under these weaker dependence conditions, we extend earlier results originally developed in the context of independent observations.