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.