Abstract:Sequential change-point detection in non-Gaussian stochastic processes is challenging because the underlying densities are rarely known. Classical parametric procedures such as CUSUM lose optimality under distributional mismatch, whereas nonparametric alternatives often react slowly. We develop a unified framework that approximates the log-likelihood ratio (LLR) on a generalized stochastic basis -- polynomial, logarithmic, or fractional-power -- using only moments up to order 3s, with no analytic form of the distribution, and thereby adapts the classical CUSUM, GRSh, and SRP procedures to non-Gaussian data. The convergence functional J(s) = K^T Y is a local Fisher-information (chi^2) measure that increases with the approximation order and converges to a ceiling fixed by the triangular (Vincze-Le Cam) discrimination, yielding a formal criterion for selecting that order. We target the regime of small relative change-points, where the signal energy changes little but the shape of the distribution -- tail structure and modality -- does. The threshold follows from Kunchenko's probability-error bound (KU-PE): a proven per-step false-alarm guarantee at s = 1, with Monte Carlo calibration of the in-control run length for s >= 2. On nine public benchmarks across four domains, the method is the only one operative on extremely heavy-tailed data (excess kurtosis gamma_4 > 20), where classical methods produce 100% false alarms, while reducing the detection delay at a controlled false-alarm rate. The core theorems are formally verified in Lean 4 (for Theorem 2, in its abstract projection form).
From: Serhii Zabolotnii Dr. [view email]
[v1]
Fri, 22 May 2026 09:28:42 UTC (202 KB)
[v2]
Wed, 27 May 2026 14:10:33 UTC (228 KB)
[v3]
Mon, 24 Aug 2026 05:22:21 UTC (170 KB)
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