Fractional Brownian Motion — Hurst Exponent & Long Memory

How the Hurst exponent H encodes long-range memory — H=0.5 is standard Brownian motion; H>0.5 is persistent (trending), H<0.5 is anti-persistent (mean-reverting)

Key insight: For fBm with Hurst exponent H, the covariance of increments is C(Δt) ∝ |t+Δt|^2H − 2|t|^2H + |t−Δt|^2H. When H=0.5 increments are uncorrelated (ordinary BM). When H>0.5, increments are positively correlated — a rise tends to continue (long memory, Nile river floods, financial volatility). When H<0.5, increments are negatively correlated — the path oscillates rapidly. H is estimated empirically via rescaled range (R/S) analysis: E[R/S] ∼ N^H.