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A constraint on extensible quadrature rules

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Here we consider integration methods with error bounded below by \(mn^{-\alpha }\) for a constant \(m>0\) and rate \(\alpha >1\). Suppose that simple averages along an extensible sequence have error at most \(Mn^{-\alpha }\) for all \(n\) in an infinite sequence of sample sizes \(n_1<n_2<\cdots \). The main result in this paper is a lower bound \(n_{k+1}/n_k\ge \rho \) where \(1<\rho <2\), so that the special sample sizes must grow at least geometrically. The bound \(\rho \) increases with \(\alpha \) and with \(m/M\). This result always rules out arithmetic sequences but never rules out sample size doubling. The same constraint holds in a root mean square error setting for any random sequence of points. Those random points need not be independent, nor uniformly distributed.

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I thank Alex Kreinin and Sergei Kucherenko for pointing out to me the elegant argument in the Appendix of [8], and Erich Novak for sharing his slides from MCQMC 2014. I also thank three anonymous reviewers for helpful suggestions. This work was supported by grants DMS-0906056 and DMS-1407397 of the U.S. National Science Foundation.

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Correspondence to Art B. Owen.

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Owen, A.B. A constraint on extensible quadrature rules. Numer. Math. 132, 511–518 (2016).

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