For the class of Gauss–Markov processes we study the problem of asymptotic equivalence of the nonparametric regression model with errors given by the increments of the process and the continuous time model, where a whole path of a sum of a deterministic signal and the Gauss–Markov process can be observed. We derive sufficient conditions which imply asymptotic equivalence of the two models. We verify these conditions for the special cases of Sobolev ellipsoids and Hölder classes with smoothness index \(>1/2\) under mild assumptions on the Gauss–Markov process. To give a counterexample, we show that asymptotic equivalence fails to hold for the special case of Brownian bridge. Our findings demonstrate that the well-known asymptotic equivalence of the Gaussian white noise model and the nonparametric regression model with i.i.d. standard normal errors (see Brown and Low (Ann Stat 24:2384–2398, 1996)) can be extended to a setup with general Gauss–Markov noises.

对于 Gauss-Markov 过程类，我们研究了非参数回归模型的渐近等价问题，误差由过程的增量和连续时间模型给出，其中确定性信号和 Gauss 之和的整个路径-可以观察到马尔科夫过程。我们推导出暗示两个模型渐近等价的充分条件。我们针对具有平滑度指数\(>1/2\)的 Sobolev 椭球和 Hölder 类的特殊情况验证了这些条件在高斯-马尔科夫过程的温和假设下。举一个反例，我们证明渐近等价对于布朗桥的特殊情况是不成立的。我们的研究结果表明，众所周知的高斯白噪声模型和具有 iid 标准正态误差的非参数回归模型的渐近等价性（参见 Brown 和 Low (Ann Stat 24:2384–2398, 1996)）可以扩展到具有一般高斯-马尔可夫噪声。