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Pré-Publication, Document De Travail Année : 2021

DISCRETE STICKY COUPLING OF FUNCTIONAL AUTOREGRESSIVE PROCESSES

Andreas Eberle
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Aurélien Enfroy
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Pierre Monmarché

Résumé

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.
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Dates et versions

hal-03199226 , version 1 (20-04-2021)

Identifiants

  • HAL Id : hal-03199226 , version 1

Citer

Alain Durmus, Andreas Eberle, Aurélien Enfroy, Arnaud Guillin, Pierre Monmarché. DISCRETE STICKY COUPLING OF FUNCTIONAL AUTOREGRESSIVE PROCESSES. 2021. ⟨hal-03199226⟩
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