Arc length asymptotics for multivariate time series

This paper quantifies the asymptotic behavior of sample arc lengths in a multivariate time series. Arc length is a natural measure of the fluctuations in a data series and can be used to quantify volatility. The idea is that processes with larger sample arc lengths exhibit larger fluctuations and hence suggest greater volatility. Here, a Gaussian functional central limit theorem for sample arc lengths is proven under finite second moment conditions. With equally spaced observations, the theory is shown to apply when the first differences of the series obey many of the popular stationary time series models in today's literature, including autoregressive moving‐average, generalized autoregressive conditional heteroscedastic, and stochastic volatility model classes. A cumulative sum statistic is introduced to identify series regimes of differing volatilities. Our applications consider log prices of asset series. Specifically, the results are used to detect nonstationary periods of stock prices. Copyright © 2014 John Wiley & Sons, Ltd.