Process monitoring has been widely recognized as an important and critical tool in system monitoring for detection of abnormal behavior and quality improvement. In manufacturing processes or industrial systems, several sources of out-of-control variation, such as tool wear, gradual equipment deterioration, vibration, inconsistent material etc., often result in dynamic changes in the process parameter, or result in special residual signals of the systems. Detecting such weak or decaying signals is a challenging task. Although the window-limited generalized likelihood ratio (wl-GLR) scheme is widely used in changepoint detection and has some advantages, it may perform poorly when it is used to detect weak or decaying signals. This paper proposes a new change-detection scheme, the window-limited weighted likelihood ratio (wl-WLR) scheme, to improve the wl-GLR scheme. To do this, a new statistical distance measure called the GLR divergence is first defined and then its properties are analyzed. The wl-WLR scheme is designed to monitor the weighted average of the GLR divergences in a moving window, and the wl-GLR scheme can be viewed as a special case of the wl-WLR scheme. Two types of weight functionals are introduced and investigated for the wl-WLR scheme. Numerical algorithms to select the optimal weight parameters are provided based on a calibration sample. Extensive simulation study favors the proposed wl-WLR scheme for detecting weak or decaying signals, and its performance is robust even if the weight parameters are not accurately estimated. This paper has online supplementary materials. (C) 2017 Elsevier Ltd. All rights reserved.