A perturbation analysis of some Markov chains models with time-varying parameters

For some families of V-geometrically ergodic Markov kernels indexed by a parameter, we study the existence of a Taylor expansion of the invariant distribution in the space of signed measures. Our approach, which completes some previous results for the perturbation analysis of Markov chains, is motivated by a problem in statistics: a control of the bias for the nonparametric kernel estimation in some locally stationary Markov models. We illustrate our results with a nonlinear autoregressive process and a Galton-Watson process with immigration and time-varying parameters.