A new version of the law of large numbers

V. N. Tutubalin

Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 1, pp. 3-17

Résumé

In this paper the system (0.1) of $N$ differential equations with random coefficients $\eta_{kj}(z)$ is considered. This system of coupled mode propagation is a mathematical model for wave-guides with random imperfections. The sum \begin{equation} \sum_{j=1}^N |E_j(z)|^2 \end{equation} is the power flow at the output of the wave-guide ($z$ is its lehgth). The physical considerations justify the investigation of an asymptotic problem when $N\to\infty$, $\eta_{kj}(z)\to 0$, $\alpha_j\to 0$, $z\to\infty$. Under some conditions the variance of the sum (1) converges to 0, while its expectation remains positive.