Asymptotic Theory for Multivariate Nonparametric Quantile Regression with Stationary Ergodic Functional Covariates and Missing-at-Random Responses

Quantiles are among the most fundamental constructs in probability theory and statistics, intrinsically linked to order structures, stochastic dominance, and the principles of robust statistical inference. Although the univariate theory of quantiles is by now classical and well developed, their generalization to multivariate settings remains mathematically subtle and methodologically demanding. In particular, extending the notion of “location within a distribution” beyond one dimension raises delicate questions of geometry, ordering, and equivariance. Within this landscape, the spatial—or geometric—formulation of multivariate quantiles has emerged as a rigorous and conceptually unifying framework capable of reconciling these issues. In this work we advance this paradigm by introducing a kernel-based estimation procedure for nonparametric conditional geometric quantiles of a multivariate response Y∈Rq (q≥2) given a functional covariate X that takes values in an infinite-dimensional space. The data are assumed to form a strictly stationary and ergodic process, while the responses may be subject to a missing-at-random mechanism, a feature of substantial practical relevance. Our analysis establishes strong consistency of the proposed estimator, characterizes its optimal convergence rate, and derives its asymptotic distribution. These limit theorems, in turn, provide the theoretical foundation for constructing asymptotically valid confidence regions and for performing inference in multivariate quantile regression with functional covariates. The theoretical developments rest on natural complexity conditions for the involved functional classes together with mild smoothness and regularity assumptions. This balance between generality and mathematical precision ensures that the resulting methodology is not only robust in a rigorous probabilistic sense but also widely applicable to contemporary problems in high-dimensional and functional data analysis. The proposed methodology is numerically investigated through simulations and is implemented in a real data application.