Aarhus University Seal / Aarhus Universitets segl

Markov Dependence in Renewal Equations and Random Sums with Heavy Tails

By Søren Asmussen and Julie Thøgersen
Thiele Research Reports
No. 02, June 2016

The Markov renewal equation \[ Z_i (x) = z_i(x) + \sum_{j \in \mathbb{E}} \int_0^x Z_j(x-y) F_{ij} (\mathrm{d} y), \qquad i \in \mathcal{E}, \] is considered in the subcritical case where the matrix of total masses of the \(F_{ij}\) has spectral radius strictly less than one, and the asymptotics of the \(Z_i(x)\) is found in the heavy-tailed case involving a local subexponential assumption on the \(F_{ij}\). Three cases occur according to the balance between the \(z_i(x)\) and the tails of the \(F_{ij}\), A crucial step in the analysis is obtaining multivariate and local versions of a lemma due to Kesten on domination of subexponential tails. These also lead to various results on tail asymptotics of sums of a random number of heavy-tailed random variables in models which are more general than in the literature.

Format available: PDF (444.3 kb)