Title: 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).