Small ball probabilities and Chung's law of the iterated logarithm for Gaussian Volterra processes with power-type kernels
arXiv:2608.05679
2026
Architecture
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper supplies a constructive family of causal Gaussian Volterra kernels whose power exponents separately control local roughness, fixed-time fluctuation scale, and global self-similar scaling. This is transferable to sequence models as a structured long-memory state or residual branch, replacing an unconstrained dense temporal filter with a low-parameter causal kernel that can interpolate between smooth, rough, and long-range-memory behavior. The most practical test is a discretized Volterra memory module with learnable exponents, compared against a same-parameter dilated convolution or SSM on long-context forecasting and language modeling.
Ideas from this paper
Unverified
2026
Add a causal memory branch whose weights are generated by the paper's power-type Volterra kernel rather than learned independently at every lag. Learn or softly constrain the exponents so the model can select rough short-memory behavior or smoother long-memory behavior while using only a few parameters. The branch can be implemented as a truncated causal convolution, a multiresolution approximation, or a recurrent state-space realization.
Useful6/10
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