Monte Carlo integration of non-differentiable functions on $[0,1]^\iota$, $\iota=1,\dots,d$ , using a single determinantal point pattern defined on $[0,1]^d$ - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Electronic Journal of Statistics Année : 2021

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

Résumé

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 $\mu(f_I)=\int_{[0,1]^\iota} f_I(u) \de u$ 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.
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hal-02508401 , version 1 (14-03-2020)

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Jean-François Coeurjolly, Adrien Mazoyer, Pierre-Olivier Amblard. Monte Carlo integration of non-differentiable functions on $[0,1]^\iota$, $\iota=1,\dots,d$ , using a single determinantal point pattern defined on $[0,1]^d$. Electronic Journal of Statistics , 2021, 15 (2), pp.6228-6280. ⟨10.1214/21-EJS1929⟩. ⟨hal-02508401⟩
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