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.
Categories: math.PR