The winding of stationary Gaussian processes

Jeremiah Buckley; Naomi Feldheim

doi:10.1007/s00440-017-0816-7

The winding of stationary Gaussian processes

Jeremiah Buckley
, Naomi Feldheim

King's College London
Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	583-614
Number of pages	32
Journal	Probability Theory and Related Fields
Volume	172
Issue number	1-2
DOIs	https://doi.org/10.1007/s00440-017-0816-7
State	Published - 1 Oct 2018
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2017, The Author(s).

Funding

Research supported by funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC Grant Agreement No 335141. Research supported in part by the United States—Israel Binational Science Foundation, Grant 2012037, and by NSF funds while the author was holding postdoctoral fellowships at the IMA Minnesota and at Stanford University. We are grateful to Mikhail Sodin for suggesting the project and for useful discussions. We thank Igor Wigman for many insightful comments, and Baruch Horovitz for a detailed conversation about the motivations coming from phyiscs. Research supported by funding from the European Research Council under the European Union?s Seventh Framework Programme (FP7/2007-2013), ERC Grant Agreement No 335141. Research supported in part by the United States?Israel Binational Science Foundation, Grant 2012037, and by NSF funds while the author was holding postdoctoral fellowships at the IMA Minnesota and at Stanford University.

Funders	Funder number
Baruch Horovitz
FP7/2007
National Science Foundation
Stanford University
Seventh Framework Programme	335141
European Commission
United States-Israel Binational Science Foundation	2012037
Seventh Framework Programme

Keywords

Fluctuations of zeroes
Gaussian process
Stationary process
Winding number

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Cite this

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The winding of stationary Gaussian processes

Abstract

Bibliographical note

Funding

Keywords

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