In this paper, we establish a new law of large numbers with the rate of convergence for special partial sums in a probability space. The proof relies on nonlinear expectation theory, as the uncertainty of random variables in the special partial sums induces the sublinearity of the expectation. As an application, we apply the new theorem to analyze the feedback channel-based detection problem of non-i.i.d. input signals in communication systems. Specifically, we investigate the convergence rates of the upper probabilities of the detection errors within the sublinear expectation space.

翻译：本文在概率空间中建立了针对特殊部分和的新的大数定律，并给出了收敛速度。该证明依赖于非线性期望理论，因为特殊部分和中随机变量的不确定性会导致期望的次线性特性。作为应用，我们运用新定理解析通信系统中基于反馈信道的非独立同分布输入信号的检测问题。具体而言，我们在次线性期望空间内研究了检测误差上概率的收敛速度。