Asymptotic normality of extensible grid sampling

Zhijian He, Lingjiong Zhu

Abstract: Recently, He and Owen (J R Stat Soc Ser B 78(4):917–931, 2016) proposed the use of Hilbert’s space filling curve (HSFC) in numerical integration as a way of reducing the dimension from $$d>1$$ d > 1 to $$d=1$$ d = 1 . This paper studies the asymptotic normality of the HSFC-based estimate when using one-dimensional stratification inputs. In particular, we are interested in using scrambled van der Corput sequence in any base $$b\ge 2$$ b ≥ 2 with sample sizes of the form $$n=b^m$$ n = b m , for which the sampling scheme is extensible in the sense of multiplying the sample size by a factor of b. We show that the estimate has an asymptotic normal distribution for functions in $$C^1([0,1]^d)$$ C 1 ( [ 0 , 1 ] d ) , excluding the trivial case of constant functions. The asymptotic normality also holds for discontinuous functions under mild conditions. Previously, it was only known that scrambled (0, m, d)-net quadratures enjoy the asymptotic normality for smooth enough functions, whose mixed partial gradients satisfy a Hölder condition. As a by-product, we find lower bounds for the variance of the HSFC-based estimate. Particularly, for non-trivial functions in $$C^1([0,1]^d)$$ C 1 ( [ 0 , 1 ] d ) , the lower bound is of order $$n^{-1-2/d}$$ n - 1 - 2 / d , which matches the rate of the upper bound established in He and Owen (2016).