Asymptotic Normality of Cumulative Cost in Linear Quadratic Regulators

The central limit theorem is a fundamental result in probability theory that characterizes the distribution of deviation from the mean in the law of large numbers. Similar distributional behavior emerges in other frameworks such as maximum likelihood estimation, least squares estimation, and stochastic approximation. In this paper, we establish a central limit theorem for the cumulative per-step cost incurred by the optimal policy in linear quadratic regulators using first principles. Our proof technique relies on a decomposition of cumulative cost using a completion of square argument, properties of the noise sequence with even density, and a central limit theorem for martingale difference sequences.