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We study the statistical inference of nonlinear stochastic approximationalgorithms utilizing a single trajectory of Markovian data. Our methodology haspractical applications in various scenarios, such as Stochastic GradientDescent (SGD) on autoregressive data and asynchronous Q-Learning. By utilizingthe standard stochastic approximation (SA) framework to estimate the targetparameter, we establish a functional central limit theorem for its partial-sumprocess, $\boldsymbol{\phi}_T$. To further support this theory, we provide amatching semiparametric efficient lower bound and a non-asymptotic upper boundon its weak convergence, measured in the L\'evy-Prokhorov metric. Thisfunctional central limit theorem forms the basis for our inference method. Byselecting any continuous scale-invariant functional $f$, the asymptotic pivotalstatistic $f(\boldsymbol{\phi}_T)$ becomes accessible, allowing us to constructan asymptotically valid confidence interval. We analyze the rejectionprobability of a family of functionals $f_m$, indexed by $m \in \mathbb{N}$,through theoretical and numerical means. The simulation results demonstrate thevalidity and efficiency of our method.

Online Statistical Inference for Nonlinear Stochastic Approximation with Markovian Data