We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random variable. To study rates of convergence for QMC on unbounded integrands, we use a smoothed projection operator to project the output of $W$ to a bounded region, which differs from the strategy of avoiding the singularities along the boundary of the unit cube $ [0,1]^d $ in 10.1137/S0036144504441573. The error is then bounded by the quadrature error of the transformed integrand and the projection error. Under certain growth conditions on the function $h$, we obtain an error rate of $ O(n^{-1+\epsilon}) $ for QMC and randomized QMC with a sample size $ n $ and an arbitrarily small $ \epsilon>0 $. Furthermore, we find that importance sampling can improve the root mean squared error of randomized QMC from $ O(n^{-1+\epsilon}) $ to $ O( n^{-3/2+\epsilon}) $.

翻译：我们考虑使用准蒙特卡罗（QMC）方法估计期望 $ \mathbb{E}\left[ h(W)\right] $ 的问题，其中 $ h $ 是 $ \mathbb{R}^d $ 上的无界光滑函数，$ W $ 是标准正态分布随机变量。为了研究QMC在无界被积函数上的收敛速率，我们采用光滑投影算子将 $W$ 的输出投影到有界区域，这与文献10.1137/S0036144504441573中避免单位立方体 $ [0,1]^d $ 边界奇异性的策略不同。误差由变换后被积函数的求积误差和投影误差共同界定。在函数 $h$ 满足一定增长条件下，我们得到QMC和随机化QMC在样本量 $ n $ 和任意小 $ \epsilon>0 $ 时的误差阶为 $ O(n^{-1+\epsilon}) $。进一步研究发现，重要性抽样可以将随机化QMC的均方根误差从 $ O(n^{-1+\epsilon}) $ 改进至 $ O( n^{-3/2+\epsilon}) $。