If you have an endless pattern of ..., -1, 1, -1, 1, -1, 1, ... and run box blur with a window of 2 or 4, you get ..., 0, 0, 0, 0, 0, 0, ... too.

Other than that, you're not wrong about theoretical Gaussian filters with infinite windows over infinite data, but this has little to do with the scenario in the article. That's about the information that leaks when you have a finite window with a discrete step and start at a well-defined boundary.