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A problem deeply investigated by multimedia forensics researchers is that of detecting which device has been used to capture a video. This enables us to trace down the owner of a video sequence, which proves extremely helpful to solve copyright infringement cases as well as to fight distribution of illicit material (e.g., child exploitation clips and terroristic threats). Currently, the most promising methods to tackle this task exploit unique noise traces left by camera sensors on acquired images. However, given the recent advancements in motion stabilization of video content, robustness of sensor pattern noise-based techniques is strongly hindered. Indeed, video stabilization introduces geometric transformations to video frames, thus making camera fingerprint estimation problematic with classical approaches. In this paper, we deal with the challenging problem of attributing stabilized videos to their recording device. Specifically, we propose: 1) a strategy to extract the characteristic fingerprint of a device, starting from either a set of images or stabilized video sequences and 2) a strategy to match a stabilized video sequence with a given fingerprint. The proposed methodology is tested on videos coming from a set of different smartphones, taken from the modern publicly available Vision Dataset. The conducted experiments also provide an interesting insight on the effect of modern smartphones video stabilization algorithms on specific video frames.