I enjoy the rare statistics textbook that can take its subject with a grain of salt:
The practitioner has heard that the [random field] should be ergodic, since “this is what makes statistical inference possible,” but is not sure how to check this fact and proceeds anyway, feeling vaguely guilty of having perhaps overlooked something very important.
–Geostatistics: Modeling Spatial Uncertainty, by Chilès and Delfiner.
It’s a familiar feeling!
As Chilès and Delfiner wryly suggest, we statisticians could often do a better job of writing for beginners or practitioners. We should not just state the assumptions needed by our tools, but also explain how sensitive results are to the assumptions, how to check these assumptions in practice, and what else to try if they’re not met.