Figure 4

Let (4.1) hold and \(\alpha_{1} = 0.3\), \(\alpha_{2} =0.3\), \(\alpha_{3} = 0.2\). Note that \(\delta_{1} = 1.1\), \(\delta_{2} = 1.05\), \(\delta_{3} = 0.65\) and \(x^{*} = G_{1}/G = 0.6641\), \(y^{*} = G_{2}/G = 0.4648\), \(z^{*} = G_{3}/G = 0.3086\), we conclude that (2.2) holds. It follows from Theorems 2.5 and 2.7 that \({\lim_{t \to + \infty} \frac{1}{t}\times}\int_{0}^{t} x (s)\,ds = \int_{\mathbb{R}_{+}^{3}} \omega_{1} \mu (d\omega_{1},d\omega_{2},d\omega_{3}) = \frac{G_{1} - \tilde{G}_{1}}{G}\) a.s., \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} y (s)\,ds = \int_{\mathbb{R}_{+}^{3}} \omega_{2} \mu (d\omega_{1}, d\omega_{2},d\omega_{3}) = \frac{G_{2} - \tilde{G}_{2}}{G}\) a.s., \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} z (s)\,ds = \int_{\mathbb{R}_{ +}^{3}} \omega_{3} \mu (d\omega_{1},d\omega_{2}, d\omega_{3}) = \frac{G_{3}- \tilde{G}_{3}}{G}\) a.s.