Fluctuations in the zeros of differentiable Gaussian processes

Area of Science:

• Mathematics
• Statistics
• Signal Processing

Background:

• Stationary Gaussian processes are fundamental in modeling random phenomena.
• Zero crossings and extremal points generate stochastic point processes.
• Autocorrelation functions characterize the statistical properties of these processes.

Purpose of the Study:

• To investigate stochastic point processes derived from Gaussian processes.
• To understand how autocorrelation function properties influence point process behavior.
• To map these properties to the parameter space of the autocorrelation function.

Main Methods:

• Analysis of point processes formed by zero crossings and extremal points.
• Studying the dependence on the autocorrelation function (ACF).
• Mapping process properties to the ACF's parameter space.

Main Results:

• Point process properties are sensitive to the ACF structure near the origin.
• Zero distributions approximate negative-binomial or binomial based on relative variance/Fano factor.
• Zeros exhibit 'antibunching' for single-scale ACFs and clustering for multi-scale ACFs.
• Interval densities become bimodal with clustering, indicating intra- and inter-cluster intervals.

Conclusions:

• The structure of the Gaussian process autocorrelation function dictates zero-crossing/extremal point process behavior.
• Clustering and antibunching phenomena are directly linked to the number of characteristic scale sizes in the ACF.
• Interevent period statistics depend on the ACF's large delay time behavior.