The Influence of Autocorrelation on Extremal Properties of Gaussian Random Processes, Correlated Randomness and Stochastic Periodicity - Ogunshemi, Alessandro (2022)
Characterising the behaviour of a random process with respect to returns to previous states is a perennial concern of stochastic process theory, with applications of results spanning the applied sciences—from reliability in physical systems and networks, to the detection of surfaces in image analysis. The Gaussian stochastic process is entirely specified by its autocorrelation, a function that measures memory or time dependence in the process and whose form can lead to changes in higher order properties of the process. For a zero-mean Gaussian process the most common returns are to the zero-level, meaning an exploration of the zero-crossings—as influenced by the autocorrelation function—is a route for studying the interplay between randomness and dependence.