Fractional Brownian Motion & Hurst Exponent

Correlated random walks with H>½ are persistent, H<½ antipersistent — memory encoded in noise.

H = 0.70
N = 512
3
Empirical 2H slope: —  |  Expected: 1.40  |  Roughness exponent: —
Fractional Brownian motion has covariance E[B_H(s)B_H(t)] = ½(s^{2H} + t^{2H} − |t−s|^{2H}).
Mean-square displacement ⟨ΔB²(τ)⟩ ~ τ^{2H}: slope on log-log gives 2H directly.
H=½ → standard Brownian motion. H>½ → persistent (trending). H<½ → antipersistent (mean-reverting).
Generated via Hosking spectral method (truncated Toeplitz).