Asymptotic Normality of Extensible Grid Sampling

Abstract

Recently, He and Owen (2016) proposed the use of Hilbert's space filling curve (HSFC) in numerical integration as a way of reducing the dimension from $d>1$ to $d=1$. This paper studies the asymptotic normality of the HSFC-based estimate when using scrambled van der Corput sequence as input. We show that the estimate has an asymptotic normal distribution for functions in $C^1([0,1]^d)$, excluding the trivial case of constant functions. The asymptotic normality also holds for discontinuous functions under mild conditions. It was previously known only 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 nontrivial functions in $C^1([0,1]^d)$, the low bound is of order $n^{-1-2/d}$, which matches the rate of the upper bound established in He and Owen (2016).