A constraint on extensible quadrature rules
A constraint on extensible quadrature rules
复制标题
可扩展求积规则的约束
作者：
A. Owen
这里我们考虑误差为$$mn^{-alpha }$$ mn-α（对于常数$$m>0$$ m>0和速率$$alpha >1$$ α>1）的积分方法。假设沿可扩展序列的简单平均在样本大小为$$n_1<n_2<cdots $$ n1<n2<⋯的无限序列中对所有$$n$$ n的误差最多为$$Mn^{-alpha }$$ Mn-α。本文的主要结果是有一个下界$$n_{k+1}/n_kge
ho $$ nk+1/nk≥ρ，其中$$1<
ho <2$$ 1<ρ<2，使得特殊样本量至少呈几何级数增长。结合的$$
ho $$ ρ随$$alpha $$ α和$$m/M$$ m/ m的增大而增大。这个结果总是排除等差序列，但从不排除样本大小加倍的可能性。同样的约束适用于任何随机点序列的均方根误差设置。这些随机点不需要是独立的，也不需要均匀分布。
Here we consider integration methods with error bounded below by $$mn^{-alpha }$$mn-α for a constant $$m>0$$m>0 and rate $$alpha >1$$α>1. Suppose that simple averages along an extensible sequence have error at most $$Mn^{-alpha }$$Mn-α for all $$n$$n in an infinite sequence of sample sizes $$n_1<n_2<cdots $$n1<n2<⋯. The main result in this paper is a lower bound $$n_{k+1}/n_kge
ho $$nk+1/nk≥ρ where $$1<
ho <2$$1<ρ<2, so that the special sample sizes must grow at least geometrically. The bound $$
ho $$ρ increases with $$alpha $$α and with $$m/M$$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.