DISCRETE STICKY COUPLING OF FUNCTIONAL AUTOREGRESSIVE PROCESSES

In this paper, we provide bounds in various metrics between the successive iterates of two functional autoregressive processes with isotropic Gaussian noise of the form $Y_{k+1}=T_\gamma(Y_k)+\sqrt{\gamma\sigma^2}Z_{k+1}$ and $\tilde Y_{k+1}=\tilde T_\gamma(\tilde Y_k)+\sqrt{\gamma\sigma^2}Z_{k+1}$ in the limit where the param- eter γ → 0. More precisely, we give non-asymptotic bounds on $\rho(L(Y_k),L(\tilde Y_k))$, where ρ is an appropriate weighted Wasserstein dis- tance or a V -distance, uniformly in the parameter γ, and on $\rho(\pi_\gamma , \tilde \pi_\gamma), where $\pi_\gamma$ and $\tilde\pi_\gamma$ are the respective stationary measures of the two processes. Of particular interest, this class of processes encompasses the Euler-Maruyama discretization of Langevin diffusions and its variants. To obtain our results, we rely on the construction of a dis- crete Markov chain $(W^{(\gamma)})_{k\in N}$ for which we are able to bound the k moments and show quantitative convergence rates uniform on γ. In addition, we show that this process converges in distribution to the continuous sticky process studied in [20, 18]. Finally, we illustrate our result on two numerical applications.