Minimax rates for Wasserstein-Kantorovich distribution deconvolution under known ordinary smooth errors and dependent signal processes

This paper addresses the problem of nonparametric deconvolution of a univariate distribution function under the Wasserstein-Kantorovich distance. We study a deconvolution model where observations are sums of a signal and an independent measurement error. The error distribution is known and ordinary smooth. The signal sequence is assumed to be strictly stationary and satisfies $\alpha$-mixing (strong mixing), $\beta$-mixing, or $\varphi$-mixing conditions--a framework that encompasses many Markov models and other dependent data structures. For example, classical ARMA processes are strongly mixing with coefficients decaying at an exponential rate. We establish non-asymptotic upper bounds on the Wasserstein risk of an approximate minimum distance kernel-based estimator of the signal distribution function in two cases: first, when no smoothness is assumed on the signal distribution; second, when the signal distribution possesses a Lebesgue density belonging to a Sobolev-type class. Matching lower bounds for the minimax risk are provided in both settings, establishing the minimax optimality of the derived convergence rates. Our results complement known findings for the i.i.d. case and represent a first step toward establishing minimax-optimal convergence rates for distribution function deconvolution under dependent signal processes.