In this paper, we focus on multiple sampling problems for the estimation of the fractional Brownian motion when the maximum number of samples is limited, extending existing results in the literature in a non-Markovian framework. Two classes of sampling schemes are proposed: a deterministic scheme and a level-triggered scheme. For the deterministic sampling scheme, the sampling times are selected beforehand and do not depend on the process trajectory. For the level-triggered sampling scheme, the sampling times are the times when the process crosses predetermined thresholds. The sampling times are selected sequentially in time and depend on the process trajectory. For each of the schemes, we derive the optimal sampling times by minimizing the aggregate squared error distortion. We then show that the optimal sampling strategies heavily depend on the dependence structure of the process.

翻译：本文在非马尔可夫框架下，研究当最大采样数量受限时，针对分数布朗运动估计的多种采样问题，拓展了现有文献中的相关结果。我们提出了两类采样方案：确定性方案和电平触发方案。对于确定性采样方案，采样时间预先选定，不依赖于过程轨迹；对于电平触发方案，采样时间为过程穿越预设阈值的时刻，采样时间按顺序动态选取，且依赖于过程轨迹。针对每种方案，我们通过最小化均方误差失真来推导最优采样时间。进而表明，最优采样策略在很大程度上取决于过程的相关结构。