We prove a sharp Gaussian approximation for the invariant law of constant-stepsize SGD with bounded additive noise generated by an exogenous uniformly ergodic Markov chain.
For a smooth, strongly convex objective with a Lipschitz Hessian and nondegenerate long-run noise covariance, the centered iterate normalized by the square root of the stepsize is $O(\sqrtα)$-close in 1-Wasserstein distance to its limiting Gaussian.
The proof combines blockwise Gaussian comparison with long-run contraction.
A four-state example gives a matching lower bound although the one-time noise marginal is symmetric and every nonzero-lag autocovariance vanishes.
In this example, an adjacent third-order mixed moment produces the leading correction.