The perturbation bound for the Perron vector of a transition probability tensor

Wen Li*, Lu Bin Cui, Kwok Po NG

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)

Abstract

In this paper, we study the perturbation bound for the Perron vector of an mth-order n-dimensional transition probability tensor P=(pi1,i2,...,im) with pi1,i2,...,im≥0 and ∑i1=1npi1,i2,...,im=1. The Perron vector x associated to the largest Z-eigenvalue 1 of P, satisfies Pxm-1=x where the entries xi of x are non-negative and ∑i=1nxi=1. The main contribution of this paper is to show that when P is perturbed to an another transition probability tensor P̃ by ΔP, the 1-norm error between x and x̃ is bounded by m, ΔP, and the computable quantity related to the uniqueness condition for the Perron vector x̃ of P̃. Based on our analysis, we can derive a new perturbation bound for the Perron vector of a transition probability matrix which refers to the case of m=2. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.

Original languageEnglish
Pages (from-to)985-1000
Number of pages16
JournalNumerical Linear Algebra with Applications
Volume20
Issue number6
DOIs
Publication statusPublished - Dec 2013

Scopus Subject Areas

  • Algebra and Number Theory
  • Applied Mathematics

User-Defined Keywords

  • Perron vector
  • Peturbation bound
  • Transition probability tensor

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