A Law of Large Numbers with Convergence Rate based on Nonlinear Expectation Theory and Its Application to Communication Detection

TL;DR

This paper develops a new law of large numbers with a convergence rate using nonlinear expectation theory, and applies it to analyze detection errors in communication systems with non-i.i.d. signals.

Contribution

It introduces a novel law of large numbers based on nonlinear expectations and demonstrates its application to communication detection problems.

Findings

Established a convergence rate for special partial sums.

Analyzed the convergence of detection error probabilities.

Applied the theorem to non-i.i.d. communication signals.

Abstract

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.