论文标题
使用单个确定点模式在$ [0,1]
Monte Carlo integration of non-differentiable functions on $[0,1]^ι$, $ι=1,\dots,d$, using a single determinantal point pattern defined on $[0,1]^d$
论文作者
论文摘要
本文涉及使用特定类别的确定点过程（DPP）（DPP）（DPP），即一类排斥的空间点过程，用于蒙特卡洛整合。令$ d \ ge 1 $，$ i \ subseteq \ edline d = \ {1，\ dots，d \} $带有$ b = | 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 $μ(f_I)=\int_{[0,1]^ι} f_I(u) \ Mathrm {d} u $对于$ [0,1]^i $上的任何已知$ b dimensional-demensable功能。特别是，我们表明所得估计器具有$ n^{ - 1-（2s \ wedge 1）/d} $的差异时，当集成剂属于具有规则性$ s> 0 $的某些Sobolev空间时。当$ s> 1/2 $（其中包括大型非差异功能）时，该方差是渐近的明确，并且显示估计器满足中心限制定理。
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 $ι=|I|$. 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 $μ(f_I)=\int_{[0,1]^ι} f_I(u) \mathrm{d} u$ for any known $ι$-dimensional integrable function on $[0,1]^ι$. 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.
