A system of nonlinear equations with application to large deviations for Markov chains with finite lifetime

Abstract

In this paper, we first show the existence of solutions to the following system of nonlinear equations \begin{eqnarray*}\left\{\begin{array}{l} a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1n}x_{n} = b_{11}\frac{1}{x_1}+b_{12}\frac{1}{x_2}+b_{13}\frac{1}{x_3}+\cdots+b_{1n}\frac{1}{x_{n}},\\ a_{21}\frac{1}{x_1}+a_{22}\frac{x_2}{x_1}+a_{23}\frac{x_3}{x_1}+\cdots+a_{2n}\frac{x_{n}}{x _1}=b_{21}x_1+b_{22}\frac{x_1}{x_2}+b_{23}\frac{x_1}{x_3}+\cdots+b_{2n}\frac{x_1}{x_{n}},\\ a_{31}\frac{x_1}{x_2}+a_{32}\frac{1}{x_2}+a_{33}\frac{x_3}{x_2}+\cdots+a_{3n}\frac{x_{n}}{x _2}=b_{31}\frac{x_2}{x_1}+b_{32}x_2+b_{33}\frac{x_2}{x_3}+\cdots+b_{3n}\frac{x_2}{x_{n}},\\ \cdots\cdots\\ a_{n1}\frac{x_1}{x_{n-1}}+a_{n2}\frac{x_2}{x_{n-1}}+a_{n3}\frac{x_3}{x_{n-1}}+ \cdots+a_{n,n-1}\frac{1}{x_{n-1}}+a_{nn}\frac{x_{n}}{x_{n-1}}\\ =b_{n1}\frac{x_{n-1}}{x_1}+b_{n2}\frac{x_{n-1}}{x_2}+b_{n3}\frac{x_{n-1}}{x_3}+\cdots+b_{n, n-1}x_{n-1} +b_{nn}\frac{x_{n-1}}{x_{n}}, \end{array} \right. \end{eqnarray*} where $n\ge 3$ and $a_{ij},b_{ij},1\le i,j\le n$, are positive constants. Then, we make use of this result to obtain the large deviation principle for the occupation time distributions of continuous-time finite state Markov chains with finite lifetime.