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.