  The Markov property is widely imposed in analysis of time series data.
Correspondingly, testing the Markov property, and relatedly, inferring the
order of a Markov model, are of paramount importance. In this article, we
propose a nonparametric test for the Markov property in high-dimensional time
series via deep conditional generative learning. We also apply the test
sequentially to determine the order of the Markov model. We show that the test
controls the type-I error asymptotically, and has the power approaching one.
Our proposal makes novel contributions in several ways. We utilize and extend
state-of-the-art deep generative learning to estimate the conditional density
functions, and establish a sharp upper bound on the approximation error of the
estimators. We derive a doubly robust test statistic, which employs a
nonparametric estimation but achieves a parametric convergence rate. We further
adopt sample splitting and cross-fitting to minimize the conditions required to
ensure the consistency of the test. We demonstrate the efficacy of the test
through both simulations and the three data applications.
