From Rare Episodes to Complete Trajectories: Prior-Preserving Diffusion for Long-Tailed Time-Series Generation

Long-tailed multivariate time-series generation is important for simulation, data sharing, and the development of reliable analysis models, yet infrequent tail states are often poorly represented in synthetic data. We identify a fundamental difficulty in window-based training, termed temporal rarity dilution: as the window length increases, the same finite tail episode appears in more highly overlapping windows without providing proportionally more independent tail information, while normal observations still dominate the learning objective. Existing time-series generators therefore struggle to preserve both sparse tail states and their temporal and multivariate context, particularly over long windows. To address this problem, we propose the Episode-to-Trajectory Diffusion Transformer (E2T-DiT), which separates when tail states occur from how their corresponding trajectories are realized. First, a duration-aware autoregressive planner models the occurrence, order, and persistence of lower-tail, normal, and upper-tail states through a compact state-duration representation. The sampled state segments are then expanded into time-aligned conditions for an Axial Diffusion Transformer, which generates complete trajectories by modeling temporal evolution and cross-variable interaction along their respective axes. Second, a prior-preserving training objective combines global denoising, state-balanced supervision, and boundary consistency, strengthening the realization of sparse tail states without altering their learned occurrence distribution. Experiments on four long-tailed time-series datasets with window lengths of 32, 64, and 128 show that E2T-DiT increases the average Correlational, Predictive, and Discriminative Scores by $3.1\%$, $10.2\%$, and $8.0\%$, respectively, compared with the strongest baseline in each setting. It achieves the best in most comparisons, and preserves its advantage as the temporal context grows.