Statistical Symmetries in Random Fields

This lecture covers the concept of statistical symmetries in random fields, focusing on stationarity and homogeneity. It explains how random fields exhibit the same statistical properties under certain transformations, such as time shifts and space shifts. The lecture delves into formal definitions and practical implications, emphasizing the invariance of moments and probability density functions. Additionally, it explores correlation functions for stationary and homogeneous random fields, highlighting the significance of these symmetries in various contexts. The lecture concludes with discussions on scaling transformations and ensemble averages in the context of random fields.