The winding of stationary Gaussian processes

• 成果类型: Article
• 署名作者: Buckley, Jeremiah; Feldheim, Naomi
• 署名单位: University of London; King's College London; Weizmann Institute of Science
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0816-7
• 发表日期: 2018
• 页码: 583-614
• 关键词: planar brownian-motion; asymptotic laws; random-walks; number; crossings; zeros

摘要

This paper studies the winding of a continuously differentiable Gaussian stationary process f : R. C in the interval [0, T]. We give formulae for the mean and the variance of this random variable. The variance is shown to always grow at least linearly with T, and conditions for it to be asymptotically linear or quadratic are given. Moreover, we show that if the covariance function together with its second derivative are in L2( R), then the winding obeys a central limit theorem. These results correspond to similar results for zeroes of real- valued stationary Gaussian functions by Malevich, Cuzick, Slud and others.