Number of Sign Changes: Segment of AR(1)

Abstract

Let $X_{t}$ denote a stationary first-order autoregressive process. Consider $n$ contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{n}$). Let its mean be zero and its lag-one serial correlation be $\rho$, which satisfies $|\rho| < 1$. Rice (1945) proved that $(n-1) \arccos(\rho)/\pi$ is the expected number of sign changes. A corresponding formula for higher-order moments was proposed by Nyberg, Lizana & Ambj\"ornsson (2018), based on an independent interval approximation. We focus on the variance only, for small $n$, and see a promising fit between theory and model.