Kernel density estimation for a stochastic process with values in a Riemannian manifold

This paper is related to the issue of the density estimation of observations with values in a Riemannian submanifold. In this context, Henry and Rodriguez ((2009), ‘Kernel Density Estimation on Riemannian Manifolds: Asymptotic Results’, Journal of Mathematical Imaging and Vision, 34, 235–239) proposed a kernel density estimator for independent data. We investigate here the behaviour of Pelletier's estimator when the observations are generated from a strictly stationary α-mixing process with values in this submanifold. Our study encompasses both pointwise and uniform analyses of the weak and strong consistency of the estimator. Specifically, we give the rate of convergence in terms of mean square error, probability, and almost sure convergence (a.s.). We also give a central-limit theorem and illustrate our proposal through some simulations and a real data application.