Is it possible to detect if the sample paths of a stochastic process almost surely admit a finite expansion with respect to some/any basis? The determination is to be made on the basis of a finite collection of discretely/noisily observed sample paths. We show that it is indeed possible to construct a hypothesis testing scheme that is almost surely guaranteed to make only finite many incorrect decisions as more data are collected. Said differently, our scheme almost certainly detects whether the process has a finite or infinite basis expansion for all sufficiently large sample sizes. Our approach relies on Cover's classical test for the irrationality of a mean, combined with tools for the non-parametric estimation of covariance operators.

是否有可能检测随机过程的样本路径是否几乎肯定承认关于某些/任何基础的有限扩展？该确定是基于离散/噪声观察样本路径的有限集合来做出的。我们证明，确实可以构建一个假设检验方案，随着收集更多数据，几乎可以肯定保证该方案只能做出有限的错误决策。换句话说，我们的方案几乎肯定会检测到该过程对于所有足够大的样本量是否具有有限或无限的基展开。我们的方法依赖于 Cover 对均值非理性的经典检验，并结合协方差算子的非参数估计工具。