Strong laws of large numbers for sequences of blockwise $m$-dependent and orthogonal random variables under sublinear expectations
Jialiang Fu
TL;DR
This work extends strong laws of large numbers to dependent sequences under sublinear expectations, focusing on blockwise $m$-dependent and orthogonal random variables. It develops capacity-based maximal inequalities, a Rademacher–Mensov-type result, and a truncation-then-limit strategy to prove SLLN-type convergence under non-additive expectations. The main contributions are Theorem 4.1 (blockwise $m$-dependent SLLN with a normalizing sequence) and Theorem 4.2 (orthogonal SLLN with a finite $ ext{E}_{ ext{hat}}[X_n^2]$ condition and a $( ef{eq})$-type summability), complemented by a quasi-orthogonal extension. Together, these results broaden limit theorems in sublinear settings, enabling robust probabilistic analysis for dependent data in uncertain environments.
Abstract
In this paper, we establish some strong laws of large numbers (SLLN) for non-independent random variables under the framework of sublinear expectations. One of our main results is for blockwise $m$-dependent random variables, and another is for orthogonal random variables. Both are the generalizations of SLLN for independent random variables in sublinear expectation spaces.