Discrete sticky couplings of functional autoregressive processes

Abstract

In this paper, we provide bounds in Wasserstein and total variation distances between the distributions of the successive iterates of two functional autoregressive processes with isotropic Gaussian noise of the form ${\mathit{Y}}_{\mathit{k}\mathbf{+}1}={\mathrm{T}}_{\mathit{\gamma }}({\mathit{Y}}_{\mathit{k}})\mathbf{+}\sqrt{\mathit{\gamma }{\mathit{\sigma }}^2}{\mathit{Z}}_{\mathit{k}\mathbf{+}1}$ and ${\tilde{\mathit{Y}}}_{\mathit{k}\mathbf{+}1}={\tilde{\mathrm{T}}}_{\mathit{\gamma }}({\tilde{\mathit{Y}}}_{\mathit{k}})\mathbf{+}\sqrt{\mathit{\gamma }{\mathit{\sigma }}^2}{\tilde{\mathit{Z}}}_{\mathit{k}\mathbf{+}1}$. More precisely, we give nonasymptotic bounds on $\mathit{\rho }(\mathcal{L}({\mathit{Y}}_{\mathit{k}}),\mathcal{L}({\tilde{\mathit{Y}}}_{\mathit{k}}))$, where ρ is an appropriate weighted Wasserstein distance or a V-distance, uniformly in the parameter γ, and on $\mathit{\rho }({\mathit{\pi }}_{\mathit{\gamma }},{\tilde{\mathit{\pi }}}_{\mathit{\gamma }})$, where ${\mathit{\pi }}_{\mathit{\gamma }}$ and ${\tilde{\mathit{\pi }}}_{\mathit{\gamma }}$ are the respective stationary measures of the two processes. The class of considered processes encompasses the Euler–Maruyama discretization of Langevin diffusions and its variants. The bounds we derive are of order γ as $\mathit{\gamma }\to 0$. To obtain our results, we rely on the construction of a discrete sticky Markov chain ${({\mathit{W}}_{\mathit{k}}^{(\mathit{\gamma })})}_{\mathit{k}\in \mathbb{N}}$ which bounds the distance between an appropriate coupling of the two processes. We then establish stability and quantitative convergence results for this process uniformly on γ. In addition, we show that it converges in distribution to the continuous sticky process studied in Howitt (Ph.D. thesis (2007)) and Eberle and Zimmer (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 2370–2394). Finally, we apply our result to Bayesian inference of ODE parameters and numerically illustrate them on two particular problems.