# Limit theorems for weighted and regular Multilevel estimators

Abstract : We aim at analyzing in terms of a.s. convergence and weak rate the performances of the Multilevel Monte Carlo estimator (MLMC) introduced in [Gil08] and of its weighted version, the Multilevel Richardson Romberg estimator (ML2R), introduced in [LP14]. These two estimators permit to compute a very accurate approximation of $I_0 = \mathbb{E}[Y_0]$ by a Monte Carlo type estimator when the (non-degenerate) random variable $Y_0 \in L^2(\mathbb{P})$ cannot be simulated (exactly) at a reasonable computational cost whereas a family of simulatable approximations $(Y_h)_{h \in \mathcal{H}}$ is available. We will carry out these investigations in an abstract framework before applying our results, mainly a Strong Law of Large Numbers and a Central Limit Theorem, to some typical fields of applications: discretization schemes of diffusions and nested Monte Carlo.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-01398292
Contributor : Vincent Lemaire <>
Submitted on : Thursday, November 17, 2016 - 9:01:47 AM
Last modification on : Saturday, March 28, 2020 - 2:09:42 AM

### Citation

Daphné Giorgi, Vincent Lemaire, Gilles Pagès. Limit theorems for weighted and regular Multilevel estimators. Monte Carlo Methods and Applications, De Gruyter, 2017, 23 (1), pp.43. ⟨10.1515/mcma-2017-0102⟩. ⟨hal-01398292⟩

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