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In this paper, we focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called Unajusted Langevin Algorithm in the Monte-Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and L 1-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using Stochastic Analysis adapted to decreasing step setting.
Consider the empirical measure, $hat{mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $mathbb{P}$ on the unit interval. For fixed $mathbb{P}$ the Wasserstein distance between $hat{mathbb{P}}_N$ and $mathbb{P}$ is a
We prove the large-dimensional Gaussian approximation of a sum of $n$ independent random vectors in $mathbb{R}^d$ together with fourth-moment error bounds on convex sets and Euclidean balls. We show that compared with classical third-moment bounds, o
We present an improved analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an addi
In this paper we give a central limit theorem for the weighted quadratic variations process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations $sum_{i=1}^{[n s]} sum_{j=1}^{[n t]} | Delta_{i,j} Y
Suppose that a random variable $X$ of interest is observed perturbed by independent additive noise $Y$. This paper concerns the the least favorable perturbation $hat Y_ep$, which maximizes the prediction error $E(X-E(X|X+Y))^2$ in the class of $Y$ wi