Number of local minima in discrete-time fractional Brownian motion

TL;DR

This paper characterizes the statistical behavior of local minima in discrete-time fractional Brownian motion, revealing a phase transition at H=3/4 between Gaussian and non-Gaussian fluctuation regimes, with implications for understanding long-range dependence.

Contribution

It provides the first complete asymptotic analysis of local minima fluctuations in non-Markovian fractional Brownian motion, identifying a phase transition at H=3/4 and linking minima count to long-range dependence diagnostics.

Findings

Fluctuations follow a Gaussian law for H ≤ 3/4.

For H > 3/4, fluctuations converge to a Rosenblatt process.

The minima count serves as a diagnostic for long-range dependence.

Abstract

The analysis of local minima in time series data and random landscapes is essential across numerous scientific disciplines, offering critical insights into system dynamics. Recently, Kundu, Majumdar, and Schehr derived the exact distribution of the number of local minima for a broad class of Markovian symmetric walks [Phys. Rev. E \textbf{110}, 024137 (2024)]; however, many real-world systems are non-Markovian, typically due to interactions with possibly hidden degrees of freedom. This work investigates the statistical properties of local minima in discrete-time samples of fractional Brownian motion (fBm), a non-Markovian Gaussian process with stationary increments, widely used to model complex, anomalous diffusion phenomena. We derive a complete asymptotic characterization of the fluctuations of the number of local minima $m_N$ in an $N$-step discrete-time fBm. We show that the fluctuations of $m_N$ exhibit a sharp transition at the Hurst exponent $H=3/4$: for $H\le 3/4$ they satisfy a central limit theorem with Gaussian limiting law, whereas for $H>3/4$ they converge to a non-Gaussian Rosenblatt process. The convergence at the process level gives us full statistical description at all times. We exemplify it on the covariance of the rescaled minima process, which displays two qualitatively distinct regimes matching Brownian and Rosenblatt covariances on either side of this threshold. Our analysis relies on a Hermite/Wick decomposition of the local-minimum indicator, which isolates a quadratic functional of an effective long-memory mode as the unique driver of the anomalous statistics. These results identify the count of local minima as a simple and robust diagnostic of long-range dependence in non-Markovian Gaussian processes, a conclusion supported by numerical simulations.