Local limit theorem for time-inhomogeneous functions of Markov processes
Leonid Koralov, Shuo Yan
TL;DR
This work proves a local limit theorem for time-inhomogeneous additive functionals $S_T=\int_0^T b(s/T,X_s)\,ds$ of a continuous-time Markov process under a quasi-compactness framework and a non-arithmetic condition. The authors develop a spectral-analytic approach based on Fourier kernels $Q(t,\alpha,\beta)$, a perturbation decomposition near $t=0$, and a careful analysis of products of these operators to extract the leading Gaussian term with covariance $\Sigma$, where $\Sigma\Sigma^*=-\int_0^1 \nabla_t^2\lambda(0,\alpha,\alpha)\,d\alpha$. They separate the analysis into regimes $t$ near and away from the origin, proving exponential decay away from zero and precise near-zero expansions, yielding the asymptotics $\prod_{k=0}^{\lfloor T\rfloor-1}\lambda(\tau/\sqrt{T},k/T,(k+1)/T) \to \exp(-\tfrac{1}{2}\tau^T\Sigma\Sigma^*\tau)$. The main result, established via an inverse Fourier transform and Breiman’s theorem, provides a local limit theorem for $S_T$ and its uniform variants, with applications to averaging in randomly perturbed linear dynamical systems; the work also clarifies the relationship between non-arithmetic and non-lattice conditions in this setting. $S_T$-level Gaussian fluctuations thus emerge from the top spectral data of the time-inhomogeneous Markov semigroup.
Abstract
In this paper, we consider a continuous-time Markov process and prove a local limit theorem for the integral of a time-inhomogeneous function of the process. One application is in the study of the fast-oscillating perturbations of linear dynamical systems.