Relaxing the stationarity assumption in time series analysis is a challenging problem. We will present an approach based on Markov chain techniques through a notion called local stationarity and which consists in replacing the usual asymptotic by an « infill » one.
The main idea is to consider time-inhomogenous Markov chains for which two successive Markov kernels are closer when the sample size increases. Such new infill asymptotics will be shown to be fundamental to give a meaning to local statistical inference when the Markov kernels depend on a functional parameter. In particular, we use contraction properties of Markov kernels to control the local approximation of the time-inhomogenous Markov chain by a stationary one. As a consequence, many parametric geometrically ergodic Markov chain models have a locally stationary version.
To recover the standard nonparametric rates of convergence for estimating functional parameters, we combine mixing properties of the Markov chains (for controlling variance terms) and differentiability properties of some families of invariant probability measures depending on a parameter (for controlling bias terms). As a consequence, we obtain a first theory for defining time-inhomogenous Markov chains models compatible with nonparametric statistical inference.