We consider the problem of detecting jumps in an otherwise smoothly evolving trend whilst the covariance and higher-order structures of the system can experience both smooth and abrupt changes over time. The number of jump points is allowed to diverge to infinity with the jump sizes possibly shrinking to zero. The method is based on a multiscale application of an optimal jump-pass filter to the time series, where the scales are dense between admissible lower and upper bounds. For a wide class of non-stationary time series models and trend functions, the proposed method is shown to be able to detect all jump points within a nearly optimal range with a prescribed probability asymptotically under mild conditions. For a time series of length $n$, the computational complexity of the proposed method is $O(n)$ for each scale and $O(n\log^{1+\epsilon} n)$ overall, where $\epsilon$ is an arbitrarily small positive constant. Numerical studies show that the proposed jump testing and estimation method performs robustly and accurately under complex temporal dynamics.

翻译：我们考虑在趋势平滑演变过程中检测跳跃点的问题，此时系统的协方差和高阶结构可能随时间发生平滑或突变。允许跳跃点数量发散至无穷大，且跳跃幅度可能趋近于零。该方法基于对时间序列应用多尺度的最优跳跃-通带滤波器，其中尺度在允许的下界与上界之间密集分布。对于一类广泛的非平稳时间序列模型与趋势函数，在温和条件下，所提方法能够以渐近给定的概率在近乎最优范围内检测出所有跳跃点。对于长度为$n$的时间序列，该方法的计算复杂度在单尺度下为$O(n)$，整体复杂度为$O(n\log^{1+\epsilon} n)$，其中$\epsilon$为任意小的正常数。数值研究表明，所提出的跳跃检验与估计方法在复杂时变动态下具有稳健且准确的表现。