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. If the function $h(\boldsymbol{x})$ and its mixed partial derivatives do not grow too fast as the Euclidean norm $|\boldsymbol{x}|$ goes to infinity, we obtain an error rate of $O(n^{-1+\epsilon})$ for QMC and randomized QMC (RQMC) with a sample size $n$ and an arbitrarily small $\epsilon>0$. However, the rate turns out to be $O(n^{-1+2M+\epsilon})$ if the functions grow exponentially with a rate of $O(\exp\{M|\boldsymbol{x}|^2\})$ for a constant $M\in(0,1/2)$. Superisingly, we find that using importance sampling with t distribution as the proposal can improve the root mean squared error of RQMC from $O(n^{-1+2M+\epsilon})$ to $O( n^{-3/2+\epsilon})$ for any $M\in(0,1/2)$.

翻译：我们考虑使用拟蒙特卡罗（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度，我们使用光滑投影算子将 $W$ 的输出投影到有界区域，这与10.1137/S0036144504441573中避免单位立方体 $[0,1]^d$ 边界奇点的策略不同。误差由变换后被积函数的求积误差和投影误差共同限定。若函数 $h(\boldsymbol{x})$ 及其混合偏导数随欧几里得范数 $|\boldsymbol{x}|$ 趋于无穷时增长不太快，当样本量为 $n$ 且 $\epsilon>0$ 任意小时，QMC和随机化QMC（RQMC）的误差率为 $O(n^{-1+\epsilon})$。然而，若函数以速率 $O(\exp\{M|\boldsymbol{x}|^2\})$ 指数增长（常数 $M\in(0,1/2)$），误差率仅为 $O(n^{-1+2M+\epsilon})$。令人惊讶的是，我们发现使用t分布作为提议分布的重要性抽样，可将RQMC的均方根误差从 $O(n^{-1+2M+\epsilon})$ 改进为 $O( n^{-3/2+\epsilon})$，该结果对任意 $M\in(0,1/2)$ 均成立。