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：在非高斯随机过程中进行序贯变点检测面临挑战，因为实时情况下底层概率密度函数鲜有已知。经典参数化方法（如CUSUM）在分布失配时会失去最优性，而非参数替代方案往往响应缓慢。我们提出一个统一框架，该框架仅利用最高至3阶矩（无需分布解析形式），在多项式、对数或分数幂的广义随机基上逼近对数似然比（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KL散度在基张成空间上的投影，由此可得选择逼近阶数的形式化准则。我们聚焦于小相对变点场景——此时信号能量变化微小，但分布形状（尾部结构与模态）发生改变。基于Kunchenko概率误差界（KU-PE）可导出稳健阈值，该阈值无需经验调参即可控制虚警率。在涵盖四个领域的九个公开基准测试中，据我们所知，本方法是唯一能在超重尾数据（超额峰度γ_4>20）上有效运行的方法——经典方法在此类数据上会产生100%虚警——同时能在保证虚警率水平下缩短检测延迟。核心定理已在Lean 4中完成形式化验证。