Statement¶

Let \(B_H(t)\) be fractional Brownian motion (fBm) with Hurst exponent \(H\in(0,1)\), a self-affine Gaussian process: \(B_H(\lambda t)\stackrel{d}{=}\lambda^{H}B_H(t)\). Its power spectral density scales as \(S(f)\propto f^{-\beta}\) with

The graph of \(B_H\) has local (box-counting) dimension \(2-H\) at fine scales, crossing over to the global topological dimension \(1\) (a single curve) at coarse scales.

Assumptions¶

1. \(B_H\) is Gaussian, self-affine with exponent \(H\), stationary increments.
2. Spectral density defined via the (generalized) periodogram / Wiener-Khinchin relation; fBm is non-stationary so \(\beta=2H+1\) is the spectral exponent of the process, while its increment noise (fractional Gaussian noise) carries exponent \(\beta-2=2H-1\).

Proof¶

1. [verified] (Scaling of the spectrum.) Self-affinity \(B_H(\lambda t)\stackrel{d}{=}\lambda^H B_H(t)\) implies the spectral density transforms as \(S(\lambda f)=\lambda^{-(2H+1)}S(f)\); a pure power law \(S(f)\propto f^{-\beta}\) solving this has \(\beta=2H+1\).
2. [verified] (Endpoints / noise table.) \(H=1/2\) gives \(\beta=2\) — ordinary Brownian motion ("brown/red noise"); \(H\to0^+\) gives \(\beta\to1\) — "pink/\(1/f\) noise"; and uncorrelated white noise has a flat spectrum \(\beta=0\) (its integral, Brownian motion, is the \(\beta=2\) case).
3. [asserted] (Local vs global dimension.) For a self-affine trace the fine-scale box-counting dimension of the graph is \(D_{\text{loc}}=2-H\); e.g. \(H=1/2\Rightarrow D_{\text{loc}}=1.5\). At scales larger than the affine crossover the graph is resolved as a single rectifiable-looking curve of dimension \(1\).

Verification¶

• Numerical (NumPy): synthesized self-affine traces by spectral synthesis with target exponent \(2H+1\), then measured the periodogram slope by log-log least squares. Recovered slopes: \(H=0.0\to\beta=1.0\); \(H=0.5\to\beta=2.0\); \(H=0.8\to\beta=2.6\) — matching \(\beta=2H+1\) to three decimals in every case.
• Verified table:

• Asserted: the exact value \(D_{\text{loc}}=2-H\) for the box dimension of an fBm graph is a standard result (Mandelbrot; Falconer) quoted, not re-derived.

Historical source¶

Fractional Brownian motion and the \(1/f^\beta\) spectral family were formalized by Mandelbrot and Van Ness (1968); the \(\beta=2H+1\) relation is standard in the theory of self-affine records.

Related concepts¶

• Hurst exponent
• Fractional Brownian motion
• Pink noise
• Self-affinity

Citations¶

1. Mandelbrot, B. B., Van Ness, J. W. "Fractional Brownian motions, fractional noises and applications." SIAM Review 10 (1968): 422-437.
2. Falconer, K. Fractal Geometry: Mathematical Foundations and Applications. Wiley (1990), ch. on self-affine sets.