This article addresses the problem of nonparametric deconvolution of a univariate cumulative distribution function with respect to the Wasserstein-Kantorovich distance. We consider a deconvolution model in which the observations consist of a signal contaminated by an independent measurement error. The error distribution is assumed to be known and ordinary smooth. The signal sequence is assumed to be strictly stationary and either \(\rho \) -mixing or \(\varphi \) -mixing (both implying \(\alpha \) -mixing), a framework that encompasses many Markov models and other dependent data structures. For instance, classical ARMA processes are \(\alpha \) -mixing (or strongly mixing) with coefficients decaying to zero at an exponential rate. We establish non-asymptotic upper bounds on the Wasserstein risk for an approximate minimum \(L^1\) -distance kernel-based estimator of the signal marginal cumulative distribution function, under the assumption that the signal has finite first absolute moment. Two distinct scenarios are analyzed: first, when no additional regularity is imposed on the signal distribution, and second, when the signal distribution has a Lebesgue density in a Sobolev-type class. In both settings, we provide matching lower bounds for the minimax risk, thereby establishing the minimax optimality of the derived convergence rates. Notably, these rates coincide with those recently established for the i.i.d. setting, indicating that weak dependence does not increase the minimax complexity of the problem. Our results complement those for the i.i.d. case and represent a first step toward establishing minimax-optimal convergence rates for cumulative distribution function deconvolution under more general dependent signal processes.
