Time-Uniform Self-Normalized Concentration for Discounted Least Squares: Limits and Corrections

Self-normalized concentration inequalities are standard tools in bandit and reinforcement-learning analyses. A widely used weighted extension claims an analogous time-uniform guarantee for discounted least-squares estimators in non-stationary problems. A simple scalar Gaussian counterexample with a fixed parameter shows that the claimed bounded radius is crossed with probability one. For fixed discount and regularization parameters, we further show that, when δ\leq1/2 and T/δ is sufficiently large, any deterministic anytime boundary valid uniformly over the stated conditionally sub-Gaussian model class must be at least of order R\sqrt{\log(T/δ)} at some time by horizon T; for nondecreasing boundaries, this order is required at time T. We identify the proof error: different terminal times use different Gaussian mixing distributions, so the fixed-time mixtures do not form one supermartingale, and the stopping-time argument does not repair this failure. Finally, we show that the weighted inequality remains valid at each fixed deterministic time, give valid finite- and infinite-horizon corrections, and discuss consequences for downstream analyses.