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Constructing a harmonic function for an irreducible nonnegative matrix with convergence parameter R >1
Author(s) -
Foley Robert D.,
McDonald David R.
Publication year - 2012
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdr115
Subject(s) - mathematics , toeplitz matrix , large deviations theory , radius of convergence , spectral radius , stochastic matrix , function (biology) , constant (computer programming) , combinatorics , harmonic function , matrix (chemical analysis) , random matrix , convergence (economics) , infinity , kernel (algebra) , discrete mathematics , pure mathematics , mathematical analysis , markov chain , eigenvalues and eigenvectors , statistics , economics , economic growth , physics , materials science , quantum mechanics , evolutionary biology , computer science , composite material , biology , programming language , power series
Let Q be a nonnegative, irreducible matrix with radius of convergence R >1. Suppose that J is a substochastic Toeplitz transition kernel J that corresponds to a random walk that is killed on the negative integers but has a positive probability of escaping to infinity. Results by Doney, J. London Math. Soc. 58 (1998) 761–768, show J has a nonnegative harmonic function U ( i ), i ∈ℕ that converges to a constant as i →∞. We show that Q has the same property, provided that Q and J are close. Note that Q is not required to be substochastic. The results in this paper are useful in understanding large deviations of a node in a queueing network that occur because of ‘cascades’, that is, because of earlier large deviations in some other part of the network that then spill over and cascade into the node of interest.