We present spectral methods of moments for learning sequential models from a single trajec-tory, in stark contrast with the classical literature that assumes the availability of multiple i.i.d. trajectories. Our approach leverages an efficient SVD-based learning algorithm for weighted au-tomata and provides the first rigorous analysis for learning many important models using dependent data. We state and analyze the algorithm under three increasingly difficult scenarios: proba-bilistic automata, stochastic weighted automata, and reactive predictive state representations controlled by a finite-state policy. Our proofs include novel tools for studying mixing properties of stochastic weighted automata.