Sequential change-point detection in non-Gaussian stochastic processes is challenging because the underlying densities are rarely known in real time. 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 interpreted as the projection of the Kullback-Leibler divergence onto the basis span, yielding a formal criterion for selecting the approximation 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. A robust threshold follows from Kunchenko's probability-error bound (KU-PE), which controls the false-alarm rate without empirical tuning. On nine public benchmarks across four domains, the method is, to our knowledge, 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 guaranteed false-alarm level. The core theorems are formally verified in Lean 4.

翻译：非高斯随机过程中的序贯变点检测极具挑战性，因为其潜在密度函数在实践中几乎无法实时获知。经典参数化方法（如CUSUM）在分布失配时会丧失最优性，而非参数替代方案往往响应迟缓。我们提出一个统一框架，该框架仅利用至多3s阶矩，无需分布的解析形式，即可在广义随机基（多项式、对数或分数幂）上近似对数似然比（LLR），从而将经典CUSUM、GRSh和SRP方法适配至非高斯数据。收敛泛函J(s)=K^T Y被解释为库尔贝克-莱布勒散度在基空间上的投影，为近似阶数的选择提供了形式化准则。我们关注小相对变点场景，此时信号能量变化甚微，但分布形态（尾部结构与模态）发生改变。基于库钦科概率误差界（KU-PE）导出的鲁棒阈值无需经验调参即可控制虚警率。在跨四个领域的九个公开基准测试中，据我们所知，该方法是唯一能在极端重尾数据（超峰度γ4>20）上正常运作的方法——经典方法在此场景下会产生100%虚警——同时能在保证虚警水平的前提下降低检测延迟。核心定理已在Lean 4中通过形式化验证。