Constructing a harmonic function for an irreducible nonnegative matrix with convergence parameter R>1

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.