An exponential inequality for widely orthant dependent random variables and its application to a first-order autoregressive model

This study develops an exponential inequality for widely orthant dependent random variables. We establish complete convergence and derive a convergence rate of $\mathcal O(1)(\log 2n)^{\frac{\alpha }{1+\alpha }}n^{-\frac{\alpha }{1+\alpha }}$ for the strong law of large numbers, where $0 \lt \alpha \leq 1$. As an application to a linear model, we obtain the strong law of large numbers with a convergence rate of $\mathcal O(1)(\log 2n)^{\frac{2\alpha }{1+\alpha }}n^{-\frac{2\alpha }{1+\alpha }}$, where $0 \lt \alpha \lt 1$. Numerical simulations are provided to illustrate and support the theoretical results.