Vol. 22, No. 2, 1967

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ISSN: 0030-8730
Ergodic properties of nonnegative matrices. I

David Vere-Jones

Vol. 22 (1967), No. 2, 361–386
Abstract

This paper contains an attempt to develop for discrete semigroups of infinite order matrices with nonnegative elements a simple theory analogous to the Perron-Frobenius theory of finite matrices. It is assumed throughout that the matrix is irreducible, but some consideration is given to the periodic case. The main topics considered are

(i) nonnegative solutions to the inequalities

 ∑
r   αikiki ≦ xi(r > 0)
k

(ii) nonnegative solutions to the inequalities

 ∑
r    xktkj ≧ xi(r > 0)
k

(iii) the limiting behaviour of sums Pj(n;r) = kuktkj(n)rn. as n →∞, where {uk} is arbitrary nonnegative vector. An extensive use is made of generating function techniques.

Mathematical Subject Classification
Primary: 60.65
Secondary: 15.00
Milestones
Received: 4 October 1965
Revised: 26 October 1966
Published: 1 August 1967
Authors
David Vere-Jones