Monte Carlo integration of non-differentiable functions on [0,1]ι, ι=1,…,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 $d\ge 1$, $I\subseteq \bar{d}=\{1,\dots ,d\}$ with $\mathrm{\iota }=|I|$. Using a single set of N quadrature points $\{u_1,\dots ,u_N\}$ defined, once for all, in dimension d from the realization of a specific DPP, we investigate “minimal” assumptions on the integrand in order to obtain unbiased Monte Carlo estimates of $\mathrm{\mu }(f_I)={\int }_{{[0,1]}^{\mathrm{\iota }}}{f}_I(u)\mathrm{d}u$ for any known ι-dimensional integrable function on ${[0,1]}^{\mathrm{\iota }}$. 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.