Number of Local Minima in Discrete-Time Fractional Brownian Motion

Area of Science:

• Statistical Physics
• Time Series Analysis
• Complex Systems

Background:

• Local minima analysis is crucial for understanding system dynamics in various scientific fields.
• Existing models often assume Markovian processes, but many real-world systems exhibit non-Markovian behavior.
• Fractional Brownian motion is a key non-Markovian model for anomalous diffusion.

Purpose of the Study:

• To investigate the statistical properties of local minima in discrete-time fractional Brownian motion.
• To characterize the fluctuations of the number of local minima in N-step samples.
• To identify the role of the Hurst exponent in determining the statistical behavior.

Main Methods:

• Derivation of asymptotic characterization for local minima fluctuations.
• Analysis using Hermite or Wick decomposition.
• Numerical simulations to support theoretical findings.

Main Results:

• A sharp transition in local minima fluctuations at Hurst exponent H=3/4.
• For H≤3/4, fluctuations follow a Central Limit Theorem with a Gaussian distribution.
• For H>3/4, fluctuations converge to a non-Gaussian Rosenblatt process.
• Identification of a quadratic functional of a long-memory mode as the driver of anomalous statistics.

Conclusions:

• The number of local minima serves as a robust diagnostic for long-range dependence in non-Markovian Gaussian processes.
• The study provides a complete statistical description of local minima fluctuations across different regimes.
• Results are validated by numerical simulations, confirming the theoretical predictions.