Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo integration using techniques from the high-dimensional statistics literature by allowing the dimension of the integral to increase. In doing so, we derive non-asymptotic bounds for the relative and absolute error of the approximation for some general classes of functions through concentration inequalities. We provide concrete examples in which the magnitude of the number of points sampled needed to guarantee a consistent estimate vary between polynomial to exponential, and show that in theory arbitrarily fast or slow rates are possible. This demonstrates that the behaviour of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain non-asymptotic confidence intervals for Monte Carlo estimate which are valid regardless of the number of points sampled.

翻译：蒙特卡洛积分是计算难以处理积分问题的常用技术，通常被认为在高维积分中表现不佳。为证明情况并非总是如此，我们借助高维统计学文献中的技术，通过允许积分维度增加来审视蒙特卡洛积分。在此过程中，利用集中不等式，针对某些一般函数类推导了近似相对误差与绝对误差的非渐近界。我们给出具体示例，其中保证一致估计所需采样点数量的量级在多项式到指数之间变化，并证明理论上可实现任意快或慢的收敛速度。这表明高维蒙特卡洛积分的表现并非单一。通过我们的方法，还获得了蒙特卡洛估计的非渐近置信区间，无论采样点数量多少，该区间均有效。