Fluctuations in the number of local minima in discrete-time fractional Brownian motion

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 expressions for the variance of the number of local minima $m_N$ in an $N$-step discrete-time fBm. In contrast to the mean of $m_N$, which depends solely on nearest-neighbor correlations, the variance captures all long-range correlations inherent in fBm. Remarkably, we find that the variance exhibits two distinct scaling regimes: for Hurst exponent $H < 3/4$, $\mathrm{Var}(m_N) \propto N$, conforming to the central limit theorem (CLT); whereas for $H > 3/4$, $\mathrm{Var}(m_N) \propto N^{4H-2}$, resulting in the breakdown of the CLT. These findings are supported by numerical simulations and provide deeper insight into the interplay between memory effects and statistical fluctuations in non-Markovian processes.