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.

翻译：马尔可夫性在时间序列数据分析中被广泛施加。相应地，检验马尔可夫性以及推断马尔可夫模型的阶数具有至关重要的意义。本文通过深度条件生成学习，提出了一种高维时间序列马尔可夫性的非参数检验方法。我们还序贯地应用该检验来确定马尔可夫模型的阶数。我们证明了该方法能渐进控制第一类错误，且检验势趋近于1。我们的方法在多个方面做出了新颖贡献。我们利用并扩展了最先进的深度生成学习来估计条件密度函数，并为估计量的逼近误差建立了尖锐的上界。我们推导了一个具有双重稳健性的检验统计量，该统计量虽采用非参数估计却达到了参数收敛速率。我们进一步采用样本分割和交叉拟合技术，以最小化确保检验一致性的所需条件。我们通过模拟实验和三项数据应用证明了该检验的有效性。