On the existence of unbiased Monte Carlo estimators
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 \(l_{2, \infty}\).
scientific article; zbMATH DE number 885467