Fractional Brownian Motion
A Gaussian stochastic process with long memory that generalizes standard Brownian motion. Its increments are self-similar and long-range correlated, and it is widely used in finance and network traffic modeling.
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Under long-memory linear processes, the convergence scale becomes γ^{β-1/2}ℓ(1/γ), and the limiting distribution is driven by an integral of fractional Brownian motion, no longer standard Gaussian.
The autocovariance of long-memory linear processes (such as FARIMA) decays slowly as a power law |k|^{2β−1}, with memory parameter β∈(1/2,1); in this case the sample mean converges at rate n^{β−1/2}, and the limiting process is the integral of fractional Brownian motion with Hurst index H=β+1/2>1/2.
All Facts (2)
Under long-memory linear processes, the convergence scale becomes γ^{β-1/2}ℓ(1/γ), and the limiting distribution is driven by an integral of fractional Brownian motion, no longer standard Gaussian.
50%UnverifiedThe autocovariance of long-memory linear processes (such as FARIMA) decays slowly as a power law |k|^{2β−1}, with memory parameter β∈(1/2,1); in this case the sample mean converges at rate n^{β−1/2}, and the limiting process is the integral of fractional Brownian motion with Hurst index H=β+1/2>1/2.
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