Fractional Brownian motions, defined as moving-average integrals of ordinary Brownian motion with a fractional-power kernel parameterized by H in (0,1), constitute self-similar Gaussian processes with stationary increments (fractional noises) whose correlations exhibit long-range dependence for H > 1/2.
The claim accurately describes fractional Brownian motion (fBm) as defined by Mandelbrot and Van Ness (1968): a moving-average integral of standard Brownian motion with a fractional-power kernel, indexed by Hurst parameter H in (0,1), yielding self-similar Gaussian processes with stationary increments. The property of long-range dependence for H > 1/2 is well-established in the literature.
Largest vulnerability: Significant uncertainty — indirect evidence is the main limit: 53% of evidence is contextual (non-decisive); 0% of sources are secondary/derivative.
The competing claim
Fractional Brownian motion need not be defined as a moving-average integral of ordinary Brownian motion; it can be equivalently and more naturally characterized via its spectral (Fourier) representation, making the moving-average kernel formulation just one of several equivalent constructions rather than the definitive definition.
The consensus frames the moving-average integral representation as the defining construction of fBm. However, spectral/Fourier-based definitions of self-similar Gaussian processes predate or parallel the Mandelbrot–Van Ness formulation. The counter-claim diverges by arguing the moving-average form is merely one representation among equals, potentially committing an appeal to authority by privileging Mandelbrot and Van Ness's 1968 formulation as canonical when mathematically equivalent alternatives exist. Mandelbrot and Van Ness themselves acknowledged spectral representations, and earlier work on random Fourier transforms supports spectral definitions as equally foundational.
Epistry weighs both sides — see the full breakdown in the app.
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