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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:459

Article

Hu, Ze-Chun²  Sun, Wei³  Zhang, Jing¹ 

[1] Hainan Normal Univ, Sch Math & Stat, Haikou 571158, Hainan, Peoples R China

[2] Sichuan Univ, Coll Math, Chengdu 610064, Sichuan, Peoples R China

[3] Concordia Univ, Dept Math & Stat, Montreal, PQ H3G 1M8, Canada

关键词: System of nonlinear equations;    Continuous-time Markov chain;    Finite lifetime;    Occupation time distribution;    Large deviation principle;   

DOI  :  10.1016/j.jmaa.2017.11.032

来源: Elsevier

【 摘 要 】

In this paper, we first show the existence of solutions to the following system of nonlinear equations {a(11)x(1) + a(12)x(2) + ... + a(1n)x(n) = b(11)1/x(1) + b(12) 1/x(2) + ... + b(1n) 1/x(n), a(21) 1/x(1) + a(22) x(2)/x(1) + ... + a(2n) x(n)/x(1) = b(21)x(1) + b(22) x(1)/x(2) + ... + b(2n) x(1)/x(n), ...... a(k,k-1) 1/x(k-1) + Sigma(1 <= j <= n j not equal k-1) a(kj) x(j)/x(k-1) = b(k,k) (-1)x(k-1) + Sigma(1 <= j <= n j not equal k-1) b(kj) x(k-1)/x(j), ...... a(n,n-1) 1/x(n-1) + Sigma(1 <= j <= n j not equal k-1) a(nj) x(j)/x(n-1) = b(n,n) (-) (1)x(n-1) + Sigma(1 <= j <= n j not equal k-1) b(nj) x(n-1)/x(j), where n >= 2 and a(jj), b(ij),1 <= i, j <= 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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free