## Abstract

The paper considers truncation errors for functions of the form f(x_{1},x_{2},…)=g(∑_{j=1} ^{∞}x_{j}ξ_{j}), i.e., errors of approximating f by f_{k}(x_{1},…,x_{k})=g(∑_{j=1} ^{k}x_{j}ξ_{j}), where the numbers ξ_{j} converge to zero sufficiently fast and x_{j}’s are i.i.d. random variables. As explained in the introduction, functions f of the form above appear in a number of important applications. To have positive results for possibly large classes of such functions, the paper provides bounds on truncation errors in both the average and worst case settings. The bounds are sharp in two out of three cases that we consider. In the former case, the functions g are from a Hilbert space G endowed with a zero mean probability measure with a given covariance kernel. In the latter case, the functions g are from a reproducing kernel Hilbert space, or a space of functions satisfying a Hölder condition.

Original language | English |
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Pages (from-to) | 52-65 |

Number of pages | 14 |

Journal | Mathematics and Computers in Simulation |

Volume | 161 |

DOIs | |

State | Published - Jul 2019 |

### Bibliographical note

Funding Information:This work was supported by the Austrian Science Fund (FWF) : Projects F5506-N26 (Kritzer) and F5509-N26 (Pillichshammer) which are parts of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

Publisher Copyright:

© 2018 International Association for Mathematics and Computers in Simulation (IMACS)

## Keywords

- Average case error
- Covariance kernel
- Dimension truncation
- Reproducing kernel
- Worst case error

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)
- Numerical Analysis
- Modeling and Simulation
- Applied Mathematics