Arc length asymptotics for multivariate time series
Article first published online: 27 MAR 2014
Copyright © 2014 John Wiley & Sons, Ltd.
Applied Stochastic Models in Business and Industry
Special Issue: ISBIS 2012 Special Issue
Volume 31, Issue 2, pages 264–281, March/April 2015
How to Cite
2015), Arc length asymptotics for multivariate time series. Appl. Stochastic Models Bus. Ind., 31: 264–281. doi: 10.1002/asmb.2030., , and (
- Issue published online: 9 APR 2015
- Article first published online: 27 MAR 2014
- Manuscript Accepted: 11 FEB 2014
- Manuscript Revised: 7 NOV 2013
- Manuscript Received: 1 SEP 2012
- central limit theorem;
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.