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 & Ambjörnsson (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.

12 pages, 1 figure

References in corpus (3)

• Zero-Crossing Statistics for Non-Markovian Time Series
• Another Look at AR(1)
• Moments of Maximum: Segment of AR(1)