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In this paper, we establish new quantitative convergence bounds for a classof functional autoregressive models in weighted total variation metrics. Toderive our results, we show that under mild assumptions, explicit minorizationand Foster-Lyapunov drift conditions hold. The main applications andconsequences of the bounds we obtain concern the geometric convergence ofEuler-Maruyama discretizations of diffusions with identity covariance matrix.Second, as a corollary, we provide a new approach to establish quantitativeconvergence of these diffusion processes by applying our conclusions in thediscrete-time setting to a well-suited sequence of discretizations whoseassociated stepsizes decrease towards zero.

Convergence of diffusions and their discretizations: from continuous to discrete processes and back