Robust Empirical Risk Minimization from a Single Dependent Trajectory: Non-Asymptotic Theory under Infinite Variance

We study robust empirical risk minimization from a single dependent trajectory, a central setting in dynamic risk management, adaptive control, and reinforcement learning. The key difficulty is the simultaneous presence of serial dependence and extreme-risk observations: dependence reduces the effective sample size, while heavy tails can cause standard empirical risk minimization to suffer from poor deviation control and high sensitivity to extreme observations. Existing robust learning theory largely treats heavy tails under independence and often bounded variance conditions, leaving a gap for modern sequential decision-making problems where observations are serially correlated and can exhibit infinite-variance behavior. We close this gap by introducing a logarithmically truncated estimator for heavy-tailed trajectory data. Under mixing and stationary conditions, we establish high-probability excess-risk guarantees that explicitly quantify the joint effects of temporal dependence and tail heaviness. The resulting rate demonstrates that blocking can recover marginal-tail-driven learning behavior up to dependence-induced logarithmic factors. As practical implications, we derive guarantees for downstream applications, including regression under autoregressive dependence and linear dynamical system identification. Numerical experiments show that the proposed estimator remains stable in regimes where standard methods break down.