We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors $r$ times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing $r$ logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings.

翻译：我们研究随机设定下加权Korobov空间中周期函数的多变量积分问题。我们提出一种新的随机秩1格则，其点数随机选取，避免在搜索优良生成向量时进行逐分量构造，同时仍能接近随机误差的最优速率。其核心思想在于：在确定设定下，至少半数可能生成向量的最坏情形误差可达到近乎最优速率。通过随机选取$r$次生成向量并比较其对应的最坏情形误差，即可高概率找到具有期望最坏情形误差界的生成向量，且（极小的）失败概率可通过令$r$随点数呈对数增长来控制。数值实验验证了理论结果。