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Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis

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Volume 12, Issue 2


A Taylor space for multivariate integration

Josef Dick

In this paper we introduce reproducing kernel Hilbert spaces based on Taylor series. The unit ball of this space contains functions which are infinite at the boundary.

We investigate multivariate integration in such spaces and show how functions in such spaces can be integrated with orderO(N −τ) for τ > 0 arbitrarily large, in spite of the unboundedness of the functions at the boundary. Further we prove that the Taylor space contains functions with infinite variance and hence the function space contains functions for which a simple Monte Carlo algorithm converges with probability one but convergence could be arbitrarily slow.

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Published Online:

Published in Print: 2006-04-01

Citation Information: Monte Carlo Methods and Applications mcma, Volume 12, Issue 2, Pages 99–112, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/156939606777488860.

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