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.

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