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This paper concerns the use of a particular class of determinantal pointprocesses (DPP), a class of repulsive spatial point processes, for Monte Carlointegration. 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 obtainunbiased 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 variancewith order $N^{-1-(2s\wedge 1)/d}$ when the integrand belongs to some Sobolevspace with regularity $s > 0$. When $s>1/2$ (which includes a large class ofnon-differentiable functions), the variance is asymptotically explicit and theestimator is shown to satisfy a Central Limit Theorem.

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