Rates of Convergence and CLTs for Subcanonical Debiased MLMC

Zheng, Zeyu; Blanchet, José

doi:10.1007/978-3-319-91436-7_26

Cited by 6 publications

(7 citation statements)

References 9 publications

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“…as t → ∞. Now, the form of the Central Limit Theorem is an immediate application of Theorem 1 in [15].…”

Section: Sketching the Proof Of Theoremmentioning

confidence: 96%

“…We have argued that, because N has finite moments of any order, the random variable N i=1 X k 2 is easily seen to have finite moments of any order. So, after applying Hölder's inequality to the right hand side of (15), it suffices to concentrate on estimating, for any q > 1,…”

Section: Sketching the Proof Of Theoremmentioning

confidence: 99%

“…Instead of an error rate of order O(1/b 1/2 ) as a function of the computational budget b, which is the typical rate, we obtain a rate of order O (log log log (b)) 1/2 /b 1/2 . The Central Limit Theorem is obtained using recently developed results in [15].…”

Section: Introductionmentioning

confidence: 99%

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Malliavin-Based Multilevel Monte Carlo Estimators for Densities of Max-Stable Processes

Blanchet

Liu

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We introduce a class of unbiased Monte Carlo estimators for the multivariate density of max-stable fields generated by Gaussian processes. Our estimators take advantage of recent results on exact simulation of max-stable fields combined with identities studied in the Malliavin calculus literature and ideas developed in the multilevel Monte Carlo literature. Our approach allows estimating multivariate densities of max-stable fields with precision ε at a computational cost of order O ε −2 log log log (1/ε) .

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“…as t → ∞. Now, the form of the Central Limit Theorem is an immediate application of Theorem 1 in [15].…”

Section: Sketching the Proof Of Theoremmentioning

confidence: 96%

Section: Sketching the Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Malliavin-Based Multilevel Monte Carlo Estimators for Densities of Max-Stable Processes

Blanchet

Liu

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…In [28,31,32] satisfying Assumption 1.1 one can construct the following unbiased coupled sampling method for the limit r.v. X:…”

Section: The Mass-shifted Mlmc Estimatormentioning

confidence: 99%

“…Although Z M clearly is not an MLMC estimator of the kind studied in this paper, one may view it, when the number of samples M is large, as a randomized MLMC estimator where both L and M ℓ ≈ M ×P (N ≥ ℓ) for all ℓ ≥ 0 are random non-negative numbers, cf. [31]. By carefully choosing the distribution of N such that Var Z i < ∞ and exploiting that Z M is the sum of i.i.d.…”

Section: The Mass-shifted Mlmc Estimatormentioning

confidence: 99%

Central limit theorems for multilevel Monte Carlo methods

Hoel

Krumscheid

2019

Journal of Complexity

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In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo (MLMC) estimators and we provide near optimal weaker conditions under which the CLT is achieved. In particular, if the variance decay rate dominates the computational cost rate (i.e., β > γ), we prove that the CLT applies to the standard (variance minimizing) MLMC estimator. For other settings where the CLT may not apply to the standard MLMC estimator, we propose an alternative estimator, called the mass-shifted MLMC estimator, to which the CLT always applies. This comes at a small efficiency loss: the computational cost of achieving mean square approximation error O(ǫ 2 ) is at worst a factor O(log(1/ǫ)) higher with the mass-shifted estimator than with the standard one.

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