Unverified50% confidenceFactExact time
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.
1
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50%
Confidence
Long-term
Relevance
10/14/2026
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Related Claims
UnverifiedIf the data exhibit long memory, using the traditional √γ confidence interval will severely underestimate uncertainty, and inference must be based on a scale of γ^{β-1/2}ℓ(1/γ).76% similarUnverifiedThe 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>1/2.76% similarUnverifiedWhen ε ≈ 1, the gradient second-moment term in the denominator is diluted, adaptive scaling has almost no effect, and the optimizer behaves close to pure SGD.67% similarUnverifiedWhen ρ has covariance Σ, the prior Gibbs distribution corresponding to the conjugate decomposition has covariance Σ^(1/3), causing the prior to also be dispersed when ρ is dispersed, resulting in better alignment between the two and preventing the Lipschitz constant from exploding.66% similarUnverifiedFor polynomial activation functions and arbitrary compact input sets, this lifting gives a rate of m^{-1/2} under the supremum norm, with an explicit constant.64% similar
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