We study the randomized n-th minimal errors (and hence the complexity) of vector valued mean computation, which is the discrete version of parametric integration. The results of the present paper form the basis for the complexity analysis of parametric integration in Sobolev spaces, which will be presented in Part 2. Altogether this extends previous results of Heinrich and Sindambiwe (1999) [12] and Wiegand (2006) [27]. Moreover, a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting is solved.

中文翻译：

---

参数积分的随机复杂性和自适应的作用 I. 有限维情况

我们研究向量值均值计算的随机第 n个最小误差（以及因此的复杂性），这是参数积分的离散版本。本文的结果构成了 Sobolev 空间参数积分复杂性分析的基础，该分析将在第 2 部分中介绍。总的来说，这扩展了 Heinrich 和 Sindambiwe (1999) [12]和 Wiegand (2006) [27的先前结果]。此外，还解决了基于信息的复杂性的基本问题，即随机设置中线性问题的自适应能力。