Overview
This research investigates the characteristics of the largest gaps observed between successive zeros of a smooth stationary Gaussian process. The central finding concerns the convergence of a rescaled joint process of these gaps to a Poisson point process, under the condition that correlations within the Gaussian process exhibit at least a polynomial decay.
Research Context
The study focuses on the behavior of zeros in stationary Gaussian processes, specifically examining the statistical properties of the largest intervals separating these zeros. The stationary nature of the process implies that its statistical properties are constant over time. The Gaussian nature refers to the process's amplitude following a Gaussian distribution. The 'smooth' characteristic suggests continuous differentiability of the process. The theoretical framework explores the implications of correlation decay rates on the extremal behavior of these gaps.
Approach
The methodology employed in this study centers on analyzing the largest gaps between successive zeros of a smooth stationary Gaussian process. A key aspect of the approach involves introducing a suitable rescaling of both the locations and sizes of these largest gaps within a growing interval. The theoretical analysis subsequently examines the resulting joint process after this rescaling. A novel step in the proof involved establishing an approximate splitting property. This property, characterized by a multiplicative error, was developed for gap events occurring in well-separated intervals. This approximate splitting property was specifically achieved for processes exhibiting arbitrarily slow polynomial decay of correlations.
The prerequisite for the main result is that the correlations within the Gaussian process must decay at least polynomially. This condition is crucial for the observed convergence behavior.
Findings
- The rescaled joint process, comprising the locations and sizes of the largest gaps between successive zeros of a smooth stationary Gaussian process, converges to a Poisson point process. This convergence is contingent on correlations decaying at least polynomially.
- A notable aspect of the proof involves establishing an approximate splitting property for gap events in well-separated intervals. This property holds with a multiplicative error.
- The approximate splitting property was demonstrated for processes where correlations decay at an arbitrarily slow polynomial rate.