Monte Carlo integration of non-differentiable functions on $[0,1]^\iota$, $\iota=1,\dots,d$, using a single determinantal point pattern defined on $[0,1]^d$

Abstract

This paper concerns the use of a particular class of determinantal point processes (DPP), a class of repulsive spatial point processes, for Monte Carlo integration. Let $d\ge 1$, $I\subseteq \overline d=\{1,\dots,d\}$ with $\iota=|I|$. Using a single set of $N$ quadrature points $\{u_1,\dots,u_N\}$ defined, once for all, in dimension $d$ from the realization of the DPP model, we investigate "minimal" assumptions on the integrand in order to obtain unbiased Monte Carlo estimates of $\mu(f_I)=\int_{[0,1]^\iota} f_I(u) \mathrm{d} u$ for any known $\iota$-dimensional integrable function on $[0,1]^\iota$. In particular, we show that the resulting estimator has variance with order $N^{-1-(2s\wedge 1)/d}$ when the integrand belongs to some Sobolev space with regularity $s > 0$. When $s>1/2$ (which includes a large class of non-differentiable functions), the variance is asymptotically explicit and the estimator is shown to satisfy a Central Limit Theorem.