On the existence of unbiased Monte Carlo estimators
type: Item
instance of: scholarly article
title: On the existence of unbiased Monte Carlo estimators
zbMATH Open document ID: 0851.65004
author: Peter Mathé
published in: Journal of Approximation Theory
publication date: 1996-11-26T00:00:00Z
review text: The author is concerned with vector-valued Monte Carlo methods for the randomized approximation of linear mappings. The question he raises is: Are there infinite-dimensional problems which admit unbiased Monte Carlo estimators? Partial answers are given indicating relations to several classes of linear operators in Banach spaces. More precisely: Def. 1. A triple \({\mathcal P}:= ([\Omega, {\mathcal F}, P], u, k)\) is called a (linear) Monte Carlo method, if: 1) \([\Omega, {\mathcal F}, P]\) is a probability space. 2) \(u: \Omega\to F(X, Y)\) is such that the mapping \(\Phi: X_0\times \Omega\to Y\), defined by \(\Phi(x, \omega):= (u(\omega))(x)\), \(x\in X\), \(\omega\in \Omega\), is product measurable in \(Y\) and the set \(\{(u(\omega))(x), x\in X, \omega\in \Omega\}\) is a separable subset in \(Y\). 3) The cardinality function \(k: \Omega\to N\) is a measurable natural number, for which \(u_\omega:= u(\omega)\in F^{k(\omega)}(X, Y)\), \(\omega\in \Omega\). Def. 2. An operator \(S\in {\mathcal L}(X, Y)\) admits an unbiased Monte Carlo method if there is a Monte Carlo method \(\mathcal P\) with: 1) \(\int_\Omega k(\omega) dP(\omega)< \infty\). 2) \(\sup_{|x|_X} \int_\Omega |\omega(x)|^2_Y dP(\omega)< \infty\). 3) \(S(x)= S_{\mathcal P} (x)\), \(x\in X\). The main result reads as: Theorem: Any Hilbert-Schmidt operator (acting between Hilbert spaces) admits an unbiased Monte Carlo method. Conversely, if an operator between Hilbert spaces admits an unbiased Monte Carlo method then necessarily the sequence of singular numbers belongs to \(l_{2, \infty}\).
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 62G99
zbMATH DE Number: 885467
zbMATH Keywords: vector valued Monte Carlo methods
zbMATH Keywords: unbiased Monte Carlo estimators
zbMATH Keywords: linear operators
zbMATH Keywords: Banach spaces
zbMATH Keywords: Hilbert-Schmidt operator
reviewed by: Jaromír Antoch
MaRDI profile type: Publication
OpenAlex ID: W1970402286
DOI: 10.1006/JATH.1996.0025
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description: scientific article; zbMATH DE number 885467