A new bivariate partial sum process for locally stationary time series is introduced and its weak convergence to a Brownian sheet is established. This construction enables the development of a novel self-normalized CUSUM test statistic for detecting changes in the mean of a locally stationary time series. For stationary data, self-normalization relies on the factorization of a constant long-run variance and a stochastic factor. In this case, the CUSUM statistic can be divided by another statistic proportional to the long-run variance, so that the latter cancels, avoiding estimation of the long-run variance. Under local stationarity, the partial sum process converges to $\int_0^t σ(x) d B_x$ and no such factorization is possible. To overcome this obstacle, a bivariate partial-sum process is introduced, allowing the construction of self-normalized test statistics under local stationarity. Weak convergence of the process is proven, and it is shown that the resulting self-normalized tests attain asymptotic level $α$ under the null hypothesis of no change, while being consistent against abrupt, gradual, and multiple changes under mild assumptions. Simulation studies show that the proposed tests have accurate size and substantially improved finite-sample power relative to existing approaches. Two data examples illustrate practical performance.

翻译：针对局部平稳时间序列，本文引入了一种新的二元部分和过程，并证明了其弱收敛于布朗片。该构造使得能够开发出一种新颖的自归一化CUSUM检验统计量，用于检测局部平稳时间序列均值的改变。对于平稳数据，自归一化依赖于常数长期方差与一个随机因子的分解。在这种情况下，CUSUM统计量可除以另一个与长期方差成比例的统计量，从而抵消长期方差，避免对其的估计。在局部平稳性条件下，部分和过程收敛于$\int_0^t σ(x) d B_x$，且无法进行此类分解。为克服这一障碍，本文引入了一种二元部分和过程，使得能够在局部平稳性下构造自归一化检验统计量。本文证明了该过程的弱收敛性，并表明所得的自归一化检验在无变化原假设下达到渐近水平$α$，同时在温和假设下对突变、渐变及多重变化具有一致性。模拟研究表明，与现有方法相比，所提出的检验具有准确的检验尺度，并显著提高了有限样本下的检验功效。两个数据实例展示了其实用性能。