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The Markov property is widely imposed in analysis of time series data.Correspondingly, testing the Markov property, and relatedly, inferring theorder of a Markov model, are of paramount importance. In this article, wepropose a nonparametric test for the Markov property in high-dimensional timeseries via deep conditional generative learning. We also apply the testsequentially to determine the order of the Markov model. We show that the testcontrols the type-I error asymptotically, and has the power approaching one.Our proposal makes novel contributions in several ways. We utilize and extendstate-of-the-art deep generative learning to estimate the conditional densityfunctions, and establish a sharp upper bound on the approximation error of theestimators. We derive a doubly robust test statistic, which employs anonparametric estimation but achieves a parametric convergence rate. We furtheradopt sample splitting and cross-fitting to minimize the conditions required toensure the consistency of the test. We demonstrate the efficacy of the testthrough both simulations and the three data applications.

Testing for the Markov Property in Time Series via Deep Conditional Generative Learning