Limit Theorems for Tempered Linear Processes with Innovations in the Domain of Attraction of a Stable Law
arXiv:2608.01674
2026
Architecture
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper identifies a precise boundary kernel between short memory and power-law long memory: a_j = exp(-lambda j) ell(j)/j, whose cumulative mass grows logarithmically through L(N) = sum_{j<=N} ell(j)/j. This suggests a causal sequence module with substantially longer memory than an exponential SSM while remaining more stable and controllable than an untempered power-law kernel. The most transferable component is the regime-aware normalization Q_N, which can prevent sequence-length and temperature-dependent activation drift when the memory scale lambda^{-1} changes.
Ideas from this paper
Unverified
2026
Replace or augment an exponential state-space memory branch with a causal convolution whose lag-j weight is exp(-lambda j) ell(j)/j. The 1/j boundary provides broad logarithmic memory, while lambda supplies an explicit finite memory scale and prevents uncontrolled accumulation from an untempered long-memory kernel.
Useful5/10
Difficulty5/10
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